Recent zbMATH articles in MSC 13https://www.zbmath.org/atom/cc/132021-02-27T13:50:00+00:00Unknown authorWerkzeugOn the projectivity of finitely generated flat modules.https://www.zbmath.org/1453.130272021-02-27T13:50:00+00:00"Tarizadeh, A."https://www.zbmath.org/authors/?q=ai:tarizadeh.abolfazlAuthor's abstract: In this paper, the projectivity of a finitely generated flat module of a commutative ring is studied through its exterior powers and invariant factors and then various new results are obtained. Specially, the related results of Endo, Vasconcelos, Wiegand, Cox-Rush and Puninski-Rothmaler on the projectivity of finitely generated flat modules are generalized.
Reviewer: François Couchot (Caen)Isolated factorizations and their applications in simplicial affine semigroups.https://www.zbmath.org/1453.200802021-02-27T13:50:00+00:00"García-Sánchez, Pedro A."https://www.zbmath.org/authors/?q=ai:garcia-sanchez.pedro-a"Herrera-Poyatos, Andrés"https://www.zbmath.org/authors/?q=ai:herrera-poyatos.andresIn this paper, the authors define the isolated factorizations in commutative monoids and some bounds for this kind of factorizations are given in numerical and simplicial affine semigroups. They use this concept in order to study the Betti elements.
They also generalize the concept of \(\alpha\)-rectangular semigroups and prove that complete intersection numerical semigroups with one Betti minimal element are \(\alpha\)-rectangular. Finally, Betti sorted semigroups and Betti divisible semigroups are presented and characterized.
A very extensive and detailed background is presented in the first section of the paper so no previous knowledge is required in order to be able to understand it.
Reviewer: Daniel Marín Aragon (Cádiz)On purely-prime ideals with applications.https://www.zbmath.org/1453.130152021-02-27T13:50:00+00:00"Tarizadeh, Abolfazl"https://www.zbmath.org/authors/?q=ai:tarizadeh.abolfazl"Aghajani, Mohsen"https://www.zbmath.org/authors/?q=ai:aghajani.mohsenSummary: In this paper, new algebraic and topological results on purely-prime ideals of a commutative ring (pure spectrum) are obtained. Particularly, Grothendieck-type theorem is obtained which states that there is a canonical correspondence between the idempotents of a ring and the clopens of its pure spectrum. It is also proved that a given ring is a Gelfand ring iff its maximal spectrum equipped with the induced Zariski topology is homeomorphic to its pure spectrum. Then as an application, it is deduced that a ring is zero dimensional iff its prime spectrum and pure spectrum are isomorphic. Dually, it is shown that a given ring is a reduced mp-ring iff its minimal spectrum equipped with the induced flat topology and its pure spectrum are the same. Finally, the new notion of semi-Noetherian ring is introduced and Cohen-type theorem is proved.Efficient relative pose estimation for cameras and generalized cameras in case of known relative rotation angle.https://www.zbmath.org/1453.130822021-02-27T13:50:00+00:00"Martyushev, Evgeniy"https://www.zbmath.org/authors/?q=ai:martyushev.evgeniy-v"Li, Bo"https://www.zbmath.org/authors/?q=ai:li.bo|li.bo.2|li.bo.1The problem of relative pose estimation of a moving camera consists in determining the current camera pose, i.e. the position and orientation of the camera, with respect to a coordinate frame related to its previous position. The standard tool for this purpose is the 5-point algorithm, which is known to be minimal, because the associated polynomial ideal is in this case zero-dimensional.
In the paper under review, the authors deal with presenting two minimal solvers to this problem for a camera with known relative rotation angle. In this directions, they describe the 4-point algorithm for regular cameras as well as the similar 5-point approach for generalized cameras. These problems are indeed formulated in terms of polynomial equations and to find the solutions of the corresponding systems, effective methods based on Gröbner bases are applied. Experiments on synthetic and real datasets show that the described algorithms are numerically efficient.
Reviewer: Amir Hashemi (Isfahan)Some properties of multiplication modules.https://www.zbmath.org/1453.130572021-02-27T13:50:00+00:00"Tavallaee, Hamid A."https://www.zbmath.org/authors/?q=ai:tavallaee.hamid-agha"Mahtabi, Robabeh"https://www.zbmath.org/authors/?q=ai:mahtabi.robabehSummary: Let \(M\) be an \(R\)-module. The module \(M\) is called multiplication if for any submodule \(N\) of \(M\) we have \(N=IM\), where \(I\) is an ideal of \(R\). In this paper we state some basic properties of submodules of these modules. Also, we study the relationship between the submodules of a multiplication \(R\)-module \(M\) and ideals of ring \(R\). Finally, by definition of semiprime submodule, we state some properties of radical submodules of multiplication modules.Super-multiplicativity of ideal norms in number fields.https://www.zbmath.org/1453.111412021-02-27T13:50:00+00:00"Marseglia, Stefano"https://www.zbmath.org/authors/?q=ai:marseglia.stefanoSummary: We study inequalities of ideal norms. We prove that in a subring \(R\) of a number field every ideal can be generated by at most three elements if and only if the ideal norm satisfies \(N(IJ)\geq N(I)N(J)\) for every pair of non-zero ideals \(I\) and \(J\) of every ring extension of \(R\) contained in the normalization \(\tilde R\).On 2-absorbing primary submodules of modules over commutative ring with unity.https://www.zbmath.org/1453.130312021-02-27T13:50:00+00:00"Dubey, Manish Kant"https://www.zbmath.org/authors/?q=ai:kant-dubey.manish"Aggarwal, Pakhi"https://www.zbmath.org/authors/?q=ai:aggarwal.pakhiThree-dimensional purely quasimonomial actions.https://www.zbmath.org/1453.120062021-02-27T13:50:00+00:00"Hoshi, Akinari"https://www.zbmath.org/authors/?q=ai:hoshi.akinari"Kitayama, Hidetaka"https://www.zbmath.org/authors/?q=ai:kitayama.hidetakaThe paper under review aims to study the rationality problem of the multiplicative actions, i.e. the \(3\)-dimensional purely quasi-monomial actions.
To begin with, we will provide an alternative definition of the purely quasi-monomial actions via \(G\)-lattices.
Let \(G\) be a finite group. Recall that a finitely generated \(\mathbb{Z}[G]\)-module \(M\) is called a \(G\)-lattice if it is torsion-free as an abelian group. As an abelian group, \(M\) becomes a free abelian group, say, of rank \(n\); the same integer \(n\) is called the rank of the \(G\)-lattice \(M\).
Let \(M=\bigoplus_{1\le i\le n} \mathbb{Z}\cdot e_i\) be a \(G\)-lattice of rank \(n\), \(K/k\) be a finite Galois extension field such that there is a surjection \(G\to \mathrm{Gal}(K/k)\). Thus \(G\) acts naturally on \(K\) by \(k\)-automorphisms. We define an action of \(G\) on \(K(M)=K(x_1,\ldots,x_n)\) the rational function field of \(n\) variables over \(K\), by \(\sigma\cdot x_j=\prod_{1\le i\le n} x_i^{a_{ij}}\) if \(\sigma\cdot e_j=\sum_{1\le i\le n} a_{ij} e_i \in M\), for any \(\sigma \in G\) (note that \(G\) acts on \(K\) as given before). The fixed field is denoted by \(K(M)^G\).
The action of \(G\) on \(K(M)\) is called a purely quasi-monomial action of \(n\) variables in this paper. It is possible that \(G\) acts faithfully on \(K\) (the case of the function field of an algebraic torus) or trivially on \(K\) (the case \(K=k\); see \textit{D. J. Saltman}'s paper [J. Algebra 133, No. 2, 533--544 (1990; Zbl 0729.13006)] and \textit{M. Hajja} and \textit{M.-C. Kang}'s papers [J. Algebra 149, No. 1, 139--154 (1992; Zbl 0760.12004); ibid. 170, No. 3, 805--860 (1994; Zbl 0831.14003)]).
The \(1\)-dimensional and the \(2\)-dimensional rationality problems of \(K(M)^G\) were solved by \textit{A. Hoshi} et al. [J. Algebra 403, 363--400 (2014; Zbl 1308.12005)]. It were shown that, \(K(M)^G\) were rational over \(k\) in some situations, and \(K(M)^G\) were rational over \(k\) if and only if some quaternion \(k\)-algebra splits in other situations. For details, see [loc. cit.] or Theorem 1.10 and Theorem 1.11 of this paper.
Thus this paper focuses on the \(3\)-dimensional case. In the main results of this paper it is assumed that char \(k \neq 2\) (this assumption was not necessary in [loc. cit.]). Since the case of algebraic tori was solved by \textit{B. È. Kunyavskiĭ} [in: Issled. Teor. Chisel, Saratov 9, 90--111 (1987; Zbl 0681.14020)], it is also assumed that the action of \(G\) on \(K\) is not faithful in Theorem 1.15 and Theorem 1.16.
Now we come to the main results of this paper, Theorem 1.15 and Theorem 1.16.
There are \(73\) finite subgroups in \(\mathrm{GL}_3(\mathbb Z)\), up to conjugation. These groups give arise to all the \(G\)-lattices of rank \(3\) (for various groups \(G\)). In the book of \textit{H. Brown} et al. [Crystallographic groups of four-dimensional space. New York etc.: John Wiley \& Sons. (1978; Zbl 0381.20002)], these groups are classified into seven crystal systems and are designated by \(G_{i,j,k}\).
The problem is solved in Theorem 1.15 if \(G\) is a group not belonging to the seventh crystal system. Theorem 1.16 takes care of the remaining cases, but the final result is incomplete. That is, if \(G\) belongs to the seventh crystal system and \(G\) is not equal to \(G_{7,j,3}\) (for \(j= 2,3,5\)), the problem is solved also. Hence there remain three groups for which the rationality problem are still unsettled.
Reviewer: Ming-Chang Kang (Taipei)Tate-Hochschild cohomology of radical square zero algebras.https://www.zbmath.org/1453.130452021-02-27T13:50:00+00:00"Wang, Zhengfang"https://www.zbmath.org/authors/?q=ai:wang.zhengfangA radical square zero algebra is such that the square of its Jacobson radical is zero. A radical square zero \(K\)-algebra is Morita equivalent to a path algebra \(KQ\) over the field \(K\) generated from a finite quiver \(Q\) modulo the ideal generated by the set of all paths of length \(2\). The Hochschild cohomology ring of radical square zero algebras were first studied by \textit{C. Cibils} [Algebras and Modules II (Geiranger, 1996), CMS Conference Proceedings, 24, 93--101 (1998)]. It was proved therein that if \(Q\) is not a \textit{crown} (a special type of quiver), the dimension of each Hochschild cohomology vector space \(\text{HH}^n(A,A)\) of a radical square zero algebra \(A\) is the difference between the number of \textit{special pairs} composed of a path of length \(n\) and an arrow sharing the same starting and ending vertices, and the number of oriented cycles of length \(n-1\). This gives the following presentation:
\[
\text{HH}^n(A,A) \cong K(Q_n// Q_1)\oplus K(Q_{n-1}// Q_0)\tag{1}
\]
where \(K(Q_n // Q_p)\) is the space generated by the set \((Q_n // Q_p)\) of special pairs \((a,b)\) of paths of length \(n\), i.e. \(a\) and length \(p\) i.e. \(b\), such that the origin vertices of \(a\) and \(b\) are the same and the terminal vertices of \(a\) and \(b\) are also the same. For example \((Q_n//Q_0)\) is the set of all oriented circles of length \(n\).
Tate-cohomology came to the fore after R. Buchweitz showed that it has connections with Maximal Cohen-Macaulay modules in [\textit{R. Buchweitz}, Manuscript, Universit\(\ddot{a}\)t Hannoover (1996)]. The singularity category \(\mathcal{D}_{sg}(A)\) of a finitely algebra \(A\) can be thought of as the full subcategory of complexes quasi-isomorphic to bounded complexes of finitely generated projective \(A\)-modules. Tate-Hochschild cohomology of \(A\) is given by \(\text{HH}^{n}_{sg}(A,A):= \text{Hom}_{\mathcal{D}_{sg}(A^e)}(A,A[n])\) where \(A^e\) is the enveloping algebra of \(A\) and \(A[n]\) is the shift functor.
The paper under review studies the Tate-Hochschild cohomology of radical square zero algebras: presenting it using combinatorial descriptions like those used in Equation (1) above. Tate-Hochschild cohomology is a Gerstenhaber algebra with the cup product being the Yoneda product in \(\mathcal{D}_{sg}(A^e)\). The author presented explicit description of the Gerstenhaber bracket on Tate-Hochschild cohomology of radical square zero algebras associated to \(c\)-crown quivers and radical square zero algebras associated to \(r\)-multiple loops quivers.
Reviewer: Tolulope Oke (College Station)A blackbox polynomial system solver on parallel shared memory computers.https://www.zbmath.org/1453.654692021-02-27T13:50:00+00:00"Verschelde, Jan"https://www.zbmath.org/authors/?q=ai:verschelde.janSummary: A numerical irreducible decomposition for a polynomial system provides representations for the irreducible factors of all positive dimensional solution sets of the system, separated from its isolated solutions. Homotopy continuation methods are applied to compute a numerical irreducible decomposition. Load balancing and pipelining are techniques in a parallel implementation on a computer with multicore processors. The application of the parallel algorithms is illustrated on solving the cyclic \(n\)-roots problems, in particular for \(n=8,9\), and \(12\).
For the entire collection see [Zbl 1396.68014].\(C(X)\): Something old and something new.https://www.zbmath.org/1453.130082021-02-27T13:50:00+00:00"Azarpanah, F."https://www.zbmath.org/authors/?q=ai:azarpanah.f"Ghashghaei, E."https://www.zbmath.org/authors/?q=ai:ghashghaei.ebrahim"Ghoulipour, M."https://www.zbmath.org/authors/?q=ai:ghoulipour.mAuthors' abstract: The aim of this paper is twofold. Firstly, to give short proofs of the celebrated results of \textit{D. Rudd} [Mich. Math. J. 17, 139--141 (1970; Zbl 0194.44403)], \textit{G. De Marco} [Proc. Am. Math. Soc. 31, 574--576 (1972; Zbl 0211.25902)], \textit{J. G. Brookshear} [Pac. J. Math. 71, 313--333 (1977; Zbl 0361.54004)], \textit{C. W. Neville} [Proc. Am. Math. Soc. 110, No. 2, 505--508 (1990; Zbl 0719.54019)], and \textit{E. M. Vechtomov} [J. Math. Sci., New York 78, No. 6, 702--753 (1996; Zbl 0868.46018)] with the help of some well-known results in ring theory. Secondly, to establish new algebraic characterizations for when \(C\)(\(X\)) is von Neumann regular, semihereditary, or a Bézout ring. The notions of the greatest common divisor and least common multiple in the ring \(C\)(\(X\)) are studied. Finally, the space X, for which every idempotent of the classical quotient ring of \(C\)(\(X\)) is actually an element of \(C\)(\(X\)) itself, is completely determined.
Reviewer: François Couchot (Caen)On \(n\)-semiprimary ideals and \(n\)-pseudo valuation domains.https://www.zbmath.org/1453.130062021-02-27T13:50:00+00:00"Anderson, David F."https://www.zbmath.org/authors/?q=ai:anderson.david-fenimore"Badawi, Ayman"https://www.zbmath.org/authors/?q=ai:badawi.aymanSummary: Let \(R\) be a commutative ring with \(1 \neq 0\) and \(n\) a positive integer. A proper ideal \(I\) of \(R\) is an \(n\)-\textit{semiprimary ideal} of \(R\) if whenever \(x^ny^n \in I\) for \(x,y \in R\), then \(x^n \in I\) or \(y^n \in I\). Let \(R\) be an integral domain with quotient field \(K\). A proper ideal \(I\) of \(R\) is an \(n\)-\textit{powerful ideal} of \(R\) if whenever \(x^ny^n \in I\) for \(x,y \in K\), then \(x^n \in R\) or \(y^n \in R\); and \(I\) is an \(n\)-\textit{powerful semiprimary ideal} of \(R\) if whenever \(x^ny^n \in I\) for \(x,y \in K\), then \(x^n \in I\) or \(y^n \in I\). If every prime ideal of \(R\) is an \(n\)-powerful semiprimary ideal of \(R\), then \(R\) is an \(n\)-\textit{pseudo-valuation domain} (\(n\)-\textit{PVD}). In this paper, we study the above concepts and relate them to several generalizations of pseudo-valuation domains.The trace map of Frobenius and extending sections for threefolds.https://www.zbmath.org/1453.140232021-02-27T13:50:00+00:00"Tanaka, Hiromu"https://www.zbmath.org/authors/?q=ai:tanaka.hiromuSummary: In this paper, by using the trace map of Frobenius, we consider problems on extending sections for positive characteristic threefolds.Multiplicative closure operations on ring extensions.https://www.zbmath.org/1453.130142021-02-27T13:50:00+00:00"Spirito, Dario"https://www.zbmath.org/authors/?q=ai:spirito.darioLet \(A \subseteq B\) be a ring extension and \( G\) be a set of \(A-\)submodules of \(B\). The author in this article introduced a new class of closure operations on \(G\) called multiplicative operations on \(A\) and \(B\), which is a generalization of the classes of star, semistar and semiprime operations.
He investigated how the set \(\mathrm{Mult}(A, B, G)\) of the multiplicative operations on \( (A, B, G)\) varies when \(A, B\) or \(G\) vary (for example, when \(A\) is integral domain and \(B\) its quotient field) and how \(\mathrm{Mult}(A, B, G)\) behaves under ring homomorphisms. Then he studied how the flexibility of the definition of this new class allows to prove functorial properties. A multiplicative operation on \((A, B, G)\) is a closure operation \( * : G \rightarrow G\), \(I \rightarrow I^*\) such that \((I : b)^* \subseteq (I^*: b)\), for all \(I \in G, b \in B \) such that \((I : b) \in G\).
Also, he show how to reduce the study of star operations on analytically unramified onedimensional Noetherian domains to the study of closures on finite extensions of Artinian rings.
Reviewer: Ismael Akray (Soran)Frobenius split subvarieties pull back in almost all characteristics.https://www.zbmath.org/1453.130232021-02-27T13:50:00+00:00"Speyer, David E."https://www.zbmath.org/authors/?q=ai:speyer.david-eSummary: Let \(\mathcal{X}\) and \(\mathcal{Y}\) be schemes of finite type over \(\operatorname{Spec} \mathbb{Z}\) and let \(\alpha : \mathcal{Y} \to \mathcal{X}\) be a finite map. We show the following holds for all sufficiently large primes \(p\): If \(\phi\) and \(\psi\) are any splittings on \(\mathcal{X}\times \operatorname{Spec} \mathbb{F}_p\) and \(\mathcal{Y} \times \operatorname{Spec} \mathbb{F}_p\), such that the restriction of \(\alpha\) is compatible with \(\phi\) and \(\psi \), and \(V\) is any compatibly split subvariety of \((\mathcal{X} \times \operatorname{Spec} \mathbb{F}_p,\phi)\), then the reduction \(\alpha^{-1} (V)^{\operatorname{red}}\) is a compatibly split subvariety of \((\mathcal{Y} \times \operatorname{Spec} \mathbb{F}_p, \psi)\). This is meant as a tool to aid in listing the compatibly split subvarieties of various classically split varieties.Subgroups of a finitary linear group.https://www.zbmath.org/1453.200712021-02-27T13:50:00+00:00"Bovdi, V."https://www.zbmath.org/authors/?q=ai:bovdi.victor-a"Dashkova, O. Yu."https://www.zbmath.org/authors/?q=ai:dashkova.olga-yu"Salim, M. A."https://www.zbmath.org/authors/?q=ai:salim.mohamed-ahmed-mSummary: Let \(\mathrm{FL}_{\nu }(K)\) be the finitary linear group of degree \(\nu\) over an associative ring \(K\) with unity. We prove that the torsion subgroups of \(\mathrm{FL}_{\nu }(K)\) are locally finite for certain classes of rings \(K\). A description of some f.g. solvable subgroups of \(\mathrm{FL}_{\nu }(K)\) are given.On the Popov-Pommerening conjecture for linear algebraic groups.https://www.zbmath.org/1453.141162021-02-27T13:50:00+00:00"Bérczi, Gergely"https://www.zbmath.org/authors/?q=ai:berczi.gergelySummary: Let \(G\) be a reductive group over an algebraically closed subfield \(k\) of \(\mathbb{C}\) of characteristic zero, \(H\subseteq G\) an observable subgroup normalised by a maximal torus of \(G\) and \(X\) an affine \(k\)-variety acted on by \(G\). Popov and Pommerening conjectured in the late 1970s that the invariant algebra \(k[X]^{H}\) is finitely generated. We prove the conjecture for: (1) subgroups of \(\mathrm{SL}_{n}(k)\) closed under left (or right) Borel action and for: (2) a class of Borel regular subgroups of classical groups. We give a partial affirmative answer to the conjecture for general regular subgroups of \(\mathrm{SL}_{n}(k)\).A computation of the Castelnuovo-Mumford regularity of certain two-dimensional unmixed ideals.https://www.zbmath.org/1453.130502021-02-27T13:50:00+00:00"Công Minh, Nguyên"https://www.zbmath.org/authors/?q=ai:nguyen-cong-minh."Thi Thuy, Phan"https://www.zbmath.org/authors/?q=ai:thuy.phan-thiLet \(K\) be a field. Let \(R\) denote the polynomial ring \(K[x_1,\dots,x_n]\) with its standard grading and let \(\mathfrak{m}:=\langle x_{1}, x_2, \dots, x_{n}\rangle\) denote its maximal homogeneous ideal. Let \(M\) be a finitely generated \(\mathbb{Z}\)-graded \(R\)-module. For each non-negative integer \(i\), it is known that \(\text{H}_{\mathfrak{m}}^i(M)\), the \(i\)th local cohomology module of \(M\) with respect to \(\mathfrak{m}\), has a natural \(\mathbb{Z}\)-grading. The \(a_i\)-invariants and Castelnuovo-Mumford regularity of \(M\) are defined by \[a_i(M):=\max \{\ell\mid\text{H}_{\mathfrak{m}}^i(M)_{\ell}\neq 0\},\quad i\in \mathbb{N}_0\] and \[\text{reg}(M):=\max \{a_i(M)+i\mid i\in
\mathbb{N}_0\}.\]
Let \(\alpha>\beta\) be natural integers and \(n\geq 5\). For every two integers \(0\leq i< j \leq n\), let \(P_{i,j}\) denote the prime ideal of \(R\) generated by the variables \(\{x_1,\dots,x_n\}\setminus \{x_i,x_j\}\). In this paper, the authors give some explicit formulas for \(a_1(R/I)\), \(a_2(R/I)\) and \(\text{reg}(R/I)\), where \(I=\bigcap\limits_{0\leq i< j\leq n} P_{i,j}^{w_{i,j}}\), \(w_{i,j}\in \{\alpha, \beta\}\).
For instance, in the case \(\alpha=\beta+1\), they show that \[a_1(R/I)=\begin{cases} \alpha+\beta-2 \ \ if \ \beta \ \text{is odd}\\
\alpha+\beta-1 \ \ if \ \beta \ \text{is even}.
\end{cases}
\]
Reviewer: Kamran Divaani-Aazar (Tehran)Enhancing the extended Hensel construction by using Gröbner bases.https://www.zbmath.org/1453.130732021-02-27T13:50:00+00:00"Sasaki, Tateaki"https://www.zbmath.org/authors/?q=ai:sasaki.tateaki"Inaba, Daiju"https://www.zbmath.org/authors/?q=ai:inaba.daijuSummary: Contrary to that the general Hensel construction (GHC) uses univariate initial Hensel factors, the extended Hensel construction (EHC) uses multivariate initial Hensel factors determined by the Newton polygon of the given multivariate polynomial. In the EHC so far, Moses-Yun's (MY) interpolation functions (see the text) are used for Hensel lifting, but the MY functions often become huge when the degree w.r.t. the main variable is large. In this paper, we propose an algorithm which uses, instead of MY functions, Gröbner bases of two initial factors which are homogeneous w.r.t. the main variable and the total-degree variable for sub-variables. The Hensel factors computed by the EHC are polynomials in the main variable with coefficients of mostly rational functions in sub-variables. We propose a method which converts the rational functions into polynomials by replacing the denominators by system variables. Each of the denominators is determined by the lowest order element of a Gröbner basis. Preliminary experiments show that our new EHC method is much faster than the previous one.
For the entire collection see [Zbl 1346.68010].On the generalized cluster algebras of geometric type.https://www.zbmath.org/1453.130632021-02-27T13:50:00+00:00"Bai, Liqian"https://www.zbmath.org/authors/?q=ai:bai.liqian"Chen, Xueqing"https://www.zbmath.org/authors/?q=ai:chen.xueqing"Ding, Ming"https://www.zbmath.org/authors/?q=ai:ding.ming"Xu, Fan"https://www.zbmath.org/authors/?q=ai:xu.fanSummary: We develop and prove the analogs of some results shown in [\textit{A. Berenstein} et al., Duke Math. J. 126, No. 1, 1--52 (2005; Zbl 1135.16013)] concerning lower and upper bounds of cluster algebras to the generalized cluster algebras of geometric type. We show that lower bounds coincide with upper bounds under the conditions of acyclicity and coprimality. Consequently, we obtain the standard monomial bases of these generalized cluster algebras. Moreover, in the appendix, we prove that an acyclic generalized cluster algebra is equal to the corresponding generalized upper cluster algebra without the assumption of the existence of coprimality.Foreword to special issue dedicated to Marco Fontana.https://www.zbmath.org/1453.000312021-02-27T13:50:00+00:00"Houston, Evan (ed.)"https://www.zbmath.org/authors/?q=ai:houston.evan-g-jun"Olberding, Bruce (ed.)"https://www.zbmath.org/authors/?q=ai:olberding.bruce-m"Salce, Luigi (ed.)"https://www.zbmath.org/authors/?q=ai:salce.luigi"Tartarone, Francesca (ed.)"https://www.zbmath.org/authors/?q=ai:tartarone.francescaFrom the text: This special issue, dedicated to the work of Marco Fontana, is a collection of original research papers representing a wide range of authors, many of whom have collaborated with Marco on past research, editorial of professional projects, and all of whom have been influenced by his far-reaching impact on commutative ring theory over the past four decades. Many of this articles reflect Marco's research interests over his long and productive career. This issue is occasioned by the conference ``Commutative Rings and their Modules, 2012'', held in Bressanone in June 2012 in honor of Marco's 65th birthday, and it is particularly fitting that such a conference should serve as the impetus for the issue, since so much progress and energy in commutative ring theory -- especially non-Noetherian commutative ring and module theory -- has resulted from the numerous conferences and publication projects organized by Marco over these last decades.Using sparse interpolation in Hensel lifting.https://www.zbmath.org/1453.130712021-02-27T13:50:00+00:00"Monagan, Michael"https://www.zbmath.org/authors/?q=ai:monagan.michael-b"Tuncer, Baris"https://www.zbmath.org/authors/?q=ai:tuncer.barisSummary: The standard approach to factor a multivariate polynomial in \(\mathbb {Z}[x_1, x_2, \ldots, x_n]\) is to factor a univariate image in \(\mathbb {Z}[x_1]\) then lift the factors of the image one variable at a time using Hensel lifting to recover the multivariate factors. At each step one must solve a multivariate polynomial Diophantine equation. For polynomials in many variables with many terms we find that solving these multivariate Diophantine equations dominates the factorization time. In this paper we explore the use of sparse interpolation methods, originally introduced by Zippel, to speed this up. We present experimental results in Maple showing that we are able to dramatically speed this up and thereby achieve a good improvement for multivariate polynomial factorization.
For the entire collection see [Zbl 1346.68010].Computing sparse representations of systems of rational fractions.https://www.zbmath.org/1453.260112021-02-27T13:50:00+00:00"Lemaire, François"https://www.zbmath.org/authors/?q=ai:lemaire.francois"Temperville, Alexandre"https://www.zbmath.org/authors/?q=ai:temperville.alexandreSummary: We present new algorithms for computing sparse representations of systems of parametric rational fractions by means of change of coordinates. Our algorithms are based on computing sparse matrices encoding the degrees of the parameters occurring in the fractions. Our methods facilitate further qualitative analysis of systems involving rational fractions. Contrary to symmetry based approaches which reduce the number of parameters, our methods only increase the sparsity, and are thus complementary. Previously hand made computations can now be fully automated by our methods.
For the entire collection see [Zbl 1346.68010].Edge ideals with regularity no more than three.https://www.zbmath.org/1453.130432021-02-27T13:50:00+00:00"Liu, Aming"https://www.zbmath.org/authors/?q=ai:liu.aming"Wu, Tongsuo"https://www.zbmath.org/authors/?q=ai:wu.tongsuoLet \(k\) be a field and \(R=k[x_1,\ldots,x_n]\) be the polynomial ring in \(n\) variables over \(k\). Suppose that \(M\) is a graded \(R\)-module with minimal free resolution
\[0 \longrightarrow \cdots \longrightarrow \bigoplus_{j}R(-j)^{\beta_{1,j}(M)} \longrightarrow \bigoplus_{j}R(-j)^{\beta_{0,j}(M)} \longrightarrow M \longrightarrow 0.\]
The Castelnuovo-Mumford regularity (or simply, regularity) of \(M\), denote by \(\mathrm{reg}(M)\), is defined as follows:
\[\mathrm{reg}(M)=\max\{j-i|\ \beta_{i,j}(M)\neq0\}.\]
There is a natural correspondence between quadratic square-free monomial ideals of \(R\) and finite simple graphs with \(n\) vertices. Indeed, to any graph \(G\), one associates its edge ideal \(I(G)\) which is generated by quadratic square-free monomials corresponding to edges of \(G\). In the paper under review, it is shown that if \(G\) is a gap-free and chair-free simple graph, then the regularity
of its edge ideal is at most \(3\).
Reviewer: S. A. Seyed Fakhari (Tehran)A note on dynamic Gröbner bases computation.https://www.zbmath.org/1453.130802021-02-27T13:50:00+00:00"Hashemi, Amir"https://www.zbmath.org/authors/?q=ai:hashemi.amir"Talaashrafi, Delaram"https://www.zbmath.org/authors/?q=ai:talaashrafi.delaramSummary: For most applications of Gröbner bases, one needs only a nice Gröbner basis of a given ideal and does not need to specify the monomial ordering. From a nice basis, we mean a basis with small size. For this purpose, \textit{P. Gritzmann} and \textit{B. Sturmfels} [SIAM J. Discrete Math. 6, No. 2, 246--269 (1993; Zbl 0798.68157)] introduced the method of dynamic Gröbner bases computation and also a variant of Buchberger's algorithm to compute a nice Gröbner basis. \textit{M. Caboara} and \textit{J. Perry} [Appl. Algebra Eng. Commun. Comput. 25, No. 1--2, 99--117 (2014; Zbl 1328.13038)] improved this approach by reducing the size and number of intermediate linear programs. In this paper, we improve the latter approach by proposing an algorithm to compute nicer Gröbner bases. The proposed algorithm has been implemented in \textsc{Sage} and its efficiency is discussed via a set of benchmark polynomials.
For the entire collection see [Zbl 1346.68010].Universal gradings of orders.https://www.zbmath.org/1453.130042021-02-27T13:50:00+00:00"Lenstra, H. W. jun."https://www.zbmath.org/authors/?q=ai:lenstra.hendrik-w-jun"Silverberg, A."https://www.zbmath.org/authors/?q=ai:silverberg.aliceSummary: For commutative rings, we introduce the notion of a \textit{universal grading}, which can be viewed as the ``largest possible grading''. While not every commutative ring (or order) has a universal grading, we prove that every \textit{reduced order} has a universal grading, and this grading is by a \textit{finite} group. Examples of graded orders are provided by group rings of finite abelian groups over rings of integers in number fields. We also generalize known properties of nilpotents, idempotents, and roots of unity in such group rings to the case of graded orders; this has applications to cryptography. Lattices play an important role in this paper; a novel aspect is that our proofs use that the additive group of any reduced order can in a natural way be equipped with a lattice structure.Direct limits of adèle rings and their completions.https://www.zbmath.org/1453.111512021-02-27T13:50:00+00:00"Kelly, James P."https://www.zbmath.org/authors/?q=ai:kelly.james-pierre"Samuels, Charles L."https://www.zbmath.org/authors/?q=ai:samuels.charles-lFor a Galois extension \(E/F\), with F a global field, the paper defines a topological ring, denoted by \(\overline{\mathbb{V}}_E\), called the generalized adèle ring of \(E\).
Denote by \(\mathcal{J}_E\) the set \(\{K \subseteq E : K/F\text{ finite Galois}\}\), by \(\mathbb{A}_K\) the adèle ring of \(K\) and by \(\overline{\mathbb{A}}_K\) its completion with respect to some (any) invariant metric on \(\mathbb{A}_K\).
Main theorems are now stated in a short form.
Theorem 1. If \(E/F\) is a Galois extension, then the following hold:
\begin{itemize}
\item[i)] \(\overline{\mathbb{V}}_E\) is a metrizable topological ring which is complete with respect to any invariant metric on \(\overline{\mathbb{V}}_E\).
\item[ii)] If \(\mathbb{V}_E = \bigcup_{K\in\mathcal J_E} \mathbb{V}_K\), then \(\overline{\mathbb{V}}_E\) equals the closure of \(\mathbb{V}_E\) in \(\overline{\mathbb{V}}_E\).
\item[iii)] There exists a topological ring isomorphism \(\phi: \overline{\mathbb{A}}_E \to \overline{\mathbb{V}}_E\) such that \(\phi(\mathbb{A}_E) = \mathbb{V}_E\).
\end{itemize}
Theorem 2.
If \(E/F\) is an infinite Galois extension, then \(\mathbb{A}_E\) has empty interior in \(\overline{\mathbb{A}}_E\).
Reviewer: Stelian Mihalas (Timişoara)The complexity of cylindrical algebraic decomposition with respect to polynomial degree.https://www.zbmath.org/1453.130792021-02-27T13:50:00+00:00"England, Matthew"https://www.zbmath.org/authors/?q=ai:england.matthew"Davenport, James H."https://www.zbmath.org/authors/?q=ai:davenport.james-haroldSummary: Cylindrical algebraic decomposition (CAD) is an important tool for working with polynomial systems, particularly quantifier elimination. However, it has complexity doubly exponential in the number of variables. The base algorithm can be improved by adapting to take advantage of any equational constraints (ECs): equations logically implied by the input. Intuitively, we expect the double exponent in the complexity to decrease by one for each EC. In ISSAC 2015 the present authors [Proceedings of the 40th international symposium on symbolic and algebraic computation. New York, NY: ACM, 165--172 (2015; Zbl 1346.68283)] proved this for the factor in the complexity bound dependent on the number of polynomials in the input. However, the other term, that dependent on the degree of the input polynomials, remained unchanged.{
} In the present paper the authors investigate how CAD in the presence of ECs could be further refined using the technology of Gröbner bases to move towards the intuitive bound for polynomial degree.
For the entire collection see [Zbl 1346.68010].A numerical method for computing border curves of bi-parametric real polynomial systems and applications.https://www.zbmath.org/1453.650392021-02-27T13:50:00+00:00"Chen, Changbo"https://www.zbmath.org/authors/?q=ai:chen.changbo"Wu, Wenyuan"https://www.zbmath.org/authors/?q=ai:wu.wenyuanSummary: For a bi-parametric real polynomial system with parameter values restricted to a finite rectangular region, under certain assumptions, we introduce the notion of border curve. We propose a numerical method to compute the border curve, and provide a numerical error estimation.{
} The border curve enables us to construct a so-called ``solution map''. For a given value \(u\) of the parameters inside the rectangle but not on the border, the solution map tells the subset that \(u\) belongs to together with a connected path from the corresponding sample point \(w\) to \(u\). Consequently, all the real solutions of the system at \(u\) (which are isolated) can be obtained by tracking a real homotopy starting from all the real roots at \(w\) throughout the path. The effectiveness of the proposed method is illustrated by some examples.
For the entire collection see [Zbl 1346.68010].Polarized Calabi-Yau 3-folds in codimension 4.https://www.zbmath.org/1453.141022021-02-27T13:50:00+00:00"Brown, Gavin"https://www.zbmath.org/authors/?q=ai:brown.gavin.2"Georgiadis, Konstantinos"https://www.zbmath.org/authors/?q=ai:georgiadis.konstantinosSummary: We construct Calabi-Yau 3-fold orbifolds embedded in weighted projective space in codimension 4. Each Hilbert series we consider is realised by at least two deformation families of Calabi-Yau 3-folds, distinguished by their topology, echoing a similar phenomenon for Fano 3-folds in high codimension.Computing all space curve solutions of polynomial systems by polyhedral methods.https://www.zbmath.org/1453.130842021-02-27T13:50:00+00:00"Bliss, Nathan"https://www.zbmath.org/authors/?q=ai:bliss.nathan"Verschelde, Jan"https://www.zbmath.org/authors/?q=ai:verschelde.janSummary: A polyhedral method to solve a system of polynomial equations exploits its sparse structure via the Newton polytopes of the polynomials. We propose a hybrid symbolic-numeric method to compute a Puiseux series expansion for every space curve that is a solution of a polynomial system. The focus of this paper concerns the difficult case when the leading powers of the Puiseux series of the space curve are contained in the relative interior of a higher dimensional cone of the tropical prevariety. We show that this difficult case does not occur for polynomials with generic coefficients. To resolve this case, we propose to apply polyhedral end games to recover tropisms hidden in the tropical prevariety.
For the entire collection see [Zbl 1346.68010].Improved computation of involutive bases.https://www.zbmath.org/1453.130742021-02-27T13:50:00+00:00"Binaei, Bentolhoda"https://www.zbmath.org/authors/?q=ai:binaei.bentolhoda"Hashemi, Amir"https://www.zbmath.org/authors/?q=ai:hashemi.amir"Seiler, Werner M."https://www.zbmath.org/authors/?q=ai:seiler.werner-mSummary: In this paper, we describe improved algorithms to compute Janet and Pommaret bases. To this end, based on the method proposed by \textit{H. Möller} et al. [International Symposium on Symbolic and Algebraic Computation, ISSAC 1992. ACM Press, 320--328 (1992; Zbl 0925.13010)], we present a more efficient variant of Gerdt's algorithm (than the algorithm presented in [\textit{V. P. Gerdt}, J. Symb. Comput. 59, 1--20 (2013; Zbl 1435.68391)]) to compute minimal involutive bases. Furthermore, by using an involutive version of the Hilbert driven technique along with the new variant of Gerdt's algorithm, we modify the algorithm given in [\textit{W. M. Seiler}, Appl. Algebra Eng. Commun. Comput. 20, No. 3-4, 261--338 (2009; Zbl 1175.13011)] to compute a linear change of coordinates for a given homogeneous ideal so that the new ideal (after performing this change) possesses a finite Pommaret basis. All the proposed algorithms have been implemented in \textsc{Maple} and their efficiency is discussed via a set of benchmark polynomials.
For the entire collection see [Zbl 1346.68010].\(F\)-pure threshold and height of quasihomogeneous polynomials.https://www.zbmath.org/1453.130222021-02-27T13:50:00+00:00"Müller, Susanne"https://www.zbmath.org/authors/?q=ai:muller.susanneSummary: The aim of this paper is to give a connection between the \(F\)-pure threshold of a polynomial and the height of the corresponding Artin-Mazur formal group. For this, we consider a quasihomogeneous polynomial \(f \in \mathbb{Z} [x_0, \ldots, x_N]\) of degree \(w\) equal to the degree of \(x_0 \cdots x_N\) and show that the \(F\)-pure threshold of the reduction \(f_p \in \mathbb{F}_p [x_0, \ldots, x_N]\) is equal to the log-canonical threshold of \(f\) if and only if the height of the Artin-Mazur formal group associated to \(H^{N - 1} (X, \mathbb{G}_{m,X})\), where \(X\) is the hypersurface given by \(f\), is equal to 1. We also prove that a similar result holds for Fermat hypersurfaces of degree greater than \(N + 1\). Furthermore, we give examples of weighted Delsarte surfaces which show that other values of the \(F\)-pure threshold of a quasihomogeneous polynomial of degree \(w\) cannot be characterized by the height.Associated primes and syzygies of linked modules.https://www.zbmath.org/1453.130462021-02-27T13:50:00+00:00"Celikbas, Olgur"https://www.zbmath.org/authors/?q=ai:celikbas.olgur"Dibaei, Mohammad T."https://www.zbmath.org/authors/?q=ai:dibaei.mohammad-t"Gheibi, Mohsen"https://www.zbmath.org/authors/?q=ai:gheibi.mohsen"Sadeghi, Arash"https://www.zbmath.org/authors/?q=ai:sadeghi.arash"Takahashi, Ryo"https://www.zbmath.org/authors/?q=ai:takahashi.ryoAs the authors mentioned in the abstract, they show that over a Gorenstein local ring \(R\), if a Cohen-Macaulay \(R\)-module \(M\) of grade \(g\) is linked to an \(R\)-module \(N\) by a Gorenstein ideal \({\mathfrak c}\) (\textit{ i.e.} \(R/{\mathfrak c}\) is a Gorenstein ring) such that \(\mathrm{Ass}_RM\) and \(\mathrm{Ass}_RN\) are disjoint, then \(M\otimes_RN\) is isomorphic to direct sum of copies of \(R/{\mathfrak a}\), where \(\mathfrak{a}\) is a Gorenstein ideal of grade \(g+1\). They also give a criterion for the depth of a local ring \((R,{\mathfrak m},k)\) in terms of the homological dimensions of the modules linked to the syzygies of the residue field \(k\). And as a result they charactrize a local ring \((R,{\mathfrak m},k)\) in terms of the homological dimensions of the modules linked to the syzygies of \(k\). For the definition of linked modules, see [\textit{A. Martsinkovsky} and \textit{J. R. Strooker}, J. Algebra 271, No. 2, 587--626 (2004; Zbl 1099.13026)].
Reviewer: Mohammad-Reza Doustimehr (Tabriz)A converse to a construction of Eisenbud-Shamash.https://www.zbmath.org/1453.130402021-02-27T13:50:00+00:00"Bergh, Petter A."https://www.zbmath.org/authors/?q=ai:bergh.petter-andreas"Jorgensen, David A."https://www.zbmath.org/authors/?q=ai:jorgensen.david-a"Moore, W. Frank"https://www.zbmath.org/authors/?q=ai:moore.w-frankSummary: Let \((Q,\mathfrak{n},k)\) be a commutative local Noetherian ring, \(f_1, \ldots, f_c\) a \(Q\)-regular sequence in \(\mathfrak{n}\), and \(R = Q/ (f_1,\ldots, f_c)\). Given a complex of finitely generated free \(R\)-modules, we give a construction of a complex of finitely generated free \(Q\)-modules having the same homology. A key application is when the original complex is an \(R\)-free resolution of a finitely generated \(R\)-module. In this case our construction is a sort of converse to a construction of \textit{D. Eisenbud} [Trans. Am. Math. Soc. 260, 35--64 (1980; Zbl 0444.13006)] which yields a free resolution of an \(R\)-module \(M\) over \(R\) given one over \(Q\).Graded Betti numbers of good filtrations.https://www.zbmath.org/1453.130172021-02-27T13:50:00+00:00"Lamei, Kamran"https://www.zbmath.org/authors/?q=ai:lamei.kamran"Yassemi, Siamak"https://www.zbmath.org/authors/?q=ai:yassemi.siamakSummary: The asymptotic behavior of graded Betti numbers of powers of homogeneous ideals in a polynomial ring over a field has recently been reviewed. We extend quasi-polynomial behavior of graded Betti numbers of powers of homogenous ideals to \(\mathbb{Z} \)-graded algebra over Noetherian local ring. Furthermore our main result treats the Betti table of filtrations which is finite or integral over the Rees algebra.Globalizing F-invariants.https://www.zbmath.org/1453.130192021-02-27T13:50:00+00:00"De Stefani, Alessandro"https://www.zbmath.org/authors/?q=ai:de-stefani.alessandro"Polstra, Thomas"https://www.zbmath.org/authors/?q=ai:polstra.thomas"Yao, Yongwei"https://www.zbmath.org/authors/?q=ai:yao.yongweiIn this nicely written paper, the authors extend the notion of the Hilbert-Kunz multiplicity and the \(F\)-signature to a non-local \(F\)-finite ring; both defined via limits which are shown to exist. As one would hope, these global versions of Hilbert-Kunz multiplicity and \(F\)-signature agree with the previously defined local versions for local \(F\)-finite local rings. They show that these new global versions satisfy the same properties that the original local versions satisfy such as:
(1) \(R\) is regular if and only if \(e_{HK}(R)=s(R)=1\).
(2) There is a positive \(\delta\) depending only on the dimension of \(R\) such that \(R\) is regular if and only if \(e_{HK}(R) \leq 1 +\delta\) if and only if \(s(R) \geq 1-\delta\).
(3) If \(R \rightarrow T\) is faithfully flat then the following inequalities hold: \(e_{HK}(R) \leq e_{HK}(T)\) and \(s(R) \geq s(T)\).
(4) \(R\) is strongly \(F\)-regular if and only if \(s(R)>0\).
(5) Let \(d\) be the Krull dimension of \(R\) and \(e(R_P)\) be the Hilbert-Samuel multiplicity of \(R_P\) and set \(e=\text{max}\{e(R_P) \mid P \in \text{Spec}(R)\}\). If \(e_{HK}(R) \leq 1 +\text{max}\{ 1/d!, 1/e\}\), then \(R\) is strongly \(F\)-regular and Gorenstein.
The authors include several examples and remarks which greatly add to the readability of the paper.
Reviewer: Janet Vassilev (Albuquerque)Corrigendum to: ``Modules satisfying the weak Nakayama property''.https://www.zbmath.org/1453.130382021-02-27T13:50:00+00:00"Samiei, M."https://www.zbmath.org/authors/?q=ai:samiei.mahdi"Moghimi, H. Fazaeli"https://www.zbmath.org/authors/?q=ai:fazaeli-moghimi.h|moghimi.hosein-fazaeliCorrects Theorem 2.7 (iii) of [\textit{M. Samiei} and \textit{H. F. Moghimi}, Indag. Math., New Ser. 25, No. 3, 553--562 (2014; Zbl 1290.13011)].On the semicontinuity problem of fibers and global \(F\)-regularity.https://www.zbmath.org/1453.130052021-02-27T13:50:00+00:00"Shimomoto, Kazuma"https://www.zbmath.org/authors/?q=ai:shimomoto.kazumaSummary: In this article, we discuss the semicontinuity problem of certain properties on fibers for a morphism of schemes. One aspect of this problem is local. Namely, we consider properties of schemes at the level of local rings, in which the main results are established by solving the lifting and localization problems for local rings. In particular, we obtain the localization theorems in the case of seminormal and \(F\)-rational rings, respectively. Another aspect of this problem is global, which is often related to the vanishing problem of certain higher direct image sheaves. As a test example, we consider the deformation of the global \(F\)-regularity.Introduction to algorithms for \(D\)-modules with quiver \(D\)-modules.https://www.zbmath.org/1453.140572021-02-27T13:50:00+00:00"Nakayama, Hiromasa"https://www.zbmath.org/authors/?q=ai:nakayama.hiromasa"Takayama, Nobuki"https://www.zbmath.org/authors/?q=ai:takayama.nobukiSummary: The goal of this expository chapter is to illustrate how to use algorithmic methods for \(D\)-modules to make mathematical experiments for \(D\)-modules and cohomology groups with examples of quiver \(D\)-modules. The first section is based on a lecture by the second author given in the Kobe-Lyon summer school 2015 On Quivers: s Computational Aspects and Geometric Applications. The second author could attend several interesting lectures of the school and the Sects. 2 and 3 are written by an inspiration from these lectures and the interesting paper by \textit{S. Khoroshkin} and \textit{A. Varchenko} [IMRP, Int. Math. Res. Pap. 2006, No. 20, Article ID 69590 116 p. (2006; Zbl 1116.32005)].
For the entire collection see [Zbl 1444.14002].On weakly semiprime subsemimodules.https://www.zbmath.org/1453.160502021-02-27T13:50:00+00:00"Farzalipour, Farkhonde"https://www.zbmath.org/authors/?q=ai:farzalipour.farkhonde"Ghiasvand, Peyman"https://www.zbmath.org/authors/?q=ai:ghiasvand.peymanSummary: In this paper we study weakly prime and weakly semiprime subsemimodules of a semimodule over a commutative semiring with nonzero identity. Also, we give a number of results concerning weakly semiprime subsemimodules of a multiplication semimodule.Corrigendum to: ``On the Cartier duality of certain finite group schemes of order \(p^n\). II.''.https://www.zbmath.org/1453.141152021-02-27T13:50:00+00:00"Amano, Michio"https://www.zbmath.org/authors/?q=ai:amano.michioSummary: We correct an error of the proof of Lemma 1 in the author's paper [ibid. 37, No. 2, 259--269 (2013; Zbl 1315.14061)]. Also a typographical error is corrected.Linear algebra for computing Gröbner bases of linear recursive multidimensional sequences.https://www.zbmath.org/1453.682222021-02-27T13:50:00+00:00"Berthomieu, Jérémy"https://www.zbmath.org/authors/?q=ai:berthomieu.jeremy"Boyer, Brice"https://www.zbmath.org/authors/?q=ai:boyer.brice"Faugère, Jean-Charles"https://www.zbmath.org/authors/?q=ai:faugere.jean-charlesSummary: The so-called Berlekamp-Massey-Sakata algorithm computes a Gröbner basis of a 0-dimensional ideal of relations satisfied by an input table. It extends the Berlekamp-Massey algorithm to \(n\)-dimensional tables, for \(n > 1\).
We investigate this problem and design several algorithms for computing such a Gröbner basis of an ideal of relations using linear algebra techniques. The first one performs a lot of table queries and is analogous to a change of variables on the ideal of relations.
As each query to the table can be expensive, we design a second algorithm requiring fewer queries, in general. This FGLM-like algorithm allows us to compute the relations of the table by extracting a full rank submatrix of a \textit{multi-Hankel} matrix (a multivariate generalization of Hankel matrices).
Under some additional assumptions, we make a third, adaptive, algorithm and reduce further the number of table queries. Then, we relate the number of queries of this third algorithm to the \textit{geometry} of the final staircase and we show that it is essentially linear in the size of the output when the staircase is convex. As a direct application to this, we decode \(n\)-cyclic codes, a generalization in dimension \(n\) of Reed Solomon codes.
We show that the multi-Hankel matrices are heavily structured when using the LEX ordering and that we can speed up the computations using fast algorithms for quasi-Hankel matrices. Finally, we design algorithms for computing the generating series of a linear recursive table.Idealization properties of comultiplication modules.https://www.zbmath.org/1453.130302021-02-27T13:50:00+00:00"Ali, Majid M."https://www.zbmath.org/authors/?q=ai:ali.majid-mSummary: In our previous work we gave a treatment of certain aspects of multiplication modules, projective modules, flat modules and like-cancellation modules via idealization. The purpose of this work is to continue our study and develop the tool of idealization in the context of comultiplication modules.On the genus of a lattice over an order of a Dedekind domain.https://www.zbmath.org/1453.130292021-02-27T13:50:00+00:00"Mba, Jules C."https://www.zbmath.org/authors/?q=ai:mba.jules-clement"Mai, Magdaline M."https://www.zbmath.org/authors/?q=ai:mai.magdaline-mSummary: The property of mutual embeddings of index not divisible by any prime in a given finite set of primes has been used successfully in the case of finitely generated groups with finite commutator subgroup to define a group structure on the non-cancellation set of such groups. If \(R\) is a Dedekind domain and \(\mathcal{O}\) is an Order over \(R\), it has been proved that lattices over \(\mathcal{O}\) belonging to the same genus have mutual embeddings. This result is formulated in this article in terms of module index and thus, allows us to define an abelian monoid structure on the genus set of such modules. We construct also some homomorphisms between genera class groups.Castelnuovo-Mumford regularity of representations of certain product categories.https://www.zbmath.org/1453.180102021-02-27T13:50:00+00:00"Gan, Wee Liang"https://www.zbmath.org/authors/?q=ai:gan.wee-liang"Li, Liping"https://www.zbmath.org/authors/?q=ai:li.lipingSummary: We show in this paper that representations of a finite product of categories satisfying certain combinatorial conditions have finite Castelnuovo-Mumford regularity if and only if they are presented in finite degrees, and hence the category consisting of them is abelian. These results apply to examples such as the categories \(\operatorname{FI}^m\) and \(\operatorname{FI}_G^m\).Recollements associated to cotorsion pairs.https://www.zbmath.org/1453.130492021-02-27T13:50:00+00:00"Chen, Wenjing"https://www.zbmath.org/authors/?q=ai:chen.wenjing"Liu, Zhongkui"https://www.zbmath.org/authors/?q=ai:liu.zhong-kui"Yang, Xiaoyan"https://www.zbmath.org/authors/?q=ai:yang.xiaoyanOn valuation independence and defectless extensions of valued fields.https://www.zbmath.org/1453.120082021-02-27T13:50:00+00:00"Cubides Kovacsics, Pablo"https://www.zbmath.org/authors/?q=ai:cubides-kovacsics.pablo"Kuhlmann, Franz-Viktor"https://www.zbmath.org/authors/?q=ai:kuhlmann.franz-viktor"Rzepka, Anna"https://www.zbmath.org/authors/?q=ai:rzepka.annaThis article develops the theory of valuation independence, a concept generalizing linear independence in the framework of vector subspaces in an extension of a valued field. Many results already existing in literature are generalized with respect to rank and dimension; moreover, they are collected in a unified form and proved relying only on algebraic methods and concept of valuation theory. The notions of \(K\)-valuation independence and \(K\)-valuation basis are introduced and strict links with the classical concepts of ``immediate extension'' and ``defectless extension'' are stated. A problem that the authors successfully face is the relation between the following two conditions:
\begin{itemize}
\item[i)] the extension \((L/K,v)\) is vs-defectless,
\item[ii)] \(L\) is linearly disjoint over \(K\) for every immediate extension of \(k\) in every common field extension;
\end{itemize}
such a problem is studied by them in a more general frame.
Reviewer: Carla Massaza (Torino)The tree of quadratic transforms of a regular local ring of dimension two.https://www.zbmath.org/1453.130112021-02-27T13:50:00+00:00"Heinzer, William"https://www.zbmath.org/authors/?q=ai:heinzer.william-j"Loper, K. Alan"https://www.zbmath.org/authors/?q=ai:loper.k-alan"Olberding, Bruce"https://www.zbmath.org/authors/?q=ai:olberding.bruce-mLet $D$ be a 2-dimensional regular local ring and $Q(D)$ denote the quadratic tree of 2-dimensional regular local overrings of $D$. They explored the topology of the tree $Q(D)$ and the family $R(D)$ of rings obtained as intersections of rings in $Q(D)$. If A is a finite intersection of rings in $Q(D)$, then $A$ is Noetherian and the structure of $A$ is well understood. However, other rings in $R(D)$ need not be Noetherian. The two main goals of this paper are to examine topological properties of the quadratic tree $Q(D)$, and to examine the
structure of rings in the set $R(D)$. For example, they shown that the patch limit points of the set $Q(D)$ in $L(D)$ is the set of all valuation rings that birationally dominate $D$. Let $V$ be a Noetherian downset in \(Q^\ast(D)\). Then for every Zariski open subset $U$ of $V$, the ring \(O_U\) is a Noetherian normal domain. Next, they showed the Zariski closure of a nonempty subset $S$ of $Q(D)$ can be calculated from how $S$ is situated in the partially ordered set \(Q^\ast(D)\) and established that every minimal valuation overring of $D$ is a localization of a ring in $R(D)$.
Reviewer: Ismael Akray (Soran)My life in mathematics, 60 years.https://www.zbmath.org/1453.130012021-02-27T13:50:00+00:00"Bokut, Leonid A."https://www.zbmath.org/authors/?q=ai:bokut.leonid-aHochster's small MCM conjecture for three-dimensional weakly F-split rings.https://www.zbmath.org/1453.130482021-02-27T13:50:00+00:00"Schoutens, Hans"https://www.zbmath.org/authors/?q=ai:schoutens.hansSummary: We prove Hochster's small maximal Cohen-Macaulay conjecture for three-dimensional complete F-pure rings. We deduce this from a more general criterion, and show that only a weakening of the notion of F-purity is needed, to wit, being weakly F-split. We conjecture that any complete ring is weakly F-split.Ulrich modules over cyclic quotient surface singularities.https://www.zbmath.org/1453.130342021-02-27T13:50:00+00:00"Nakajima, Yusuke"https://www.zbmath.org/authors/?q=ai:nakajima.yusuke"Yoshida, Ken-Ichi"https://www.zbmath.org/authors/?q=ai:yoshida.ken-ichi|yoshida.ken-ichi.2Summary: In this paper, we characterize Ulrich modules over cyclic quotient surface singularities using the notion of special Cohen-Macaulay modules. We also investigate the number of indecomposable Ulrich modules for a given cyclic quotient surface singularity, and show that the number of exceptional curves in the minimal resolution determines a boundary on the number of indecomposable Ulrich modules.Atomic and AP semigroup rings \(F[X;M]\), where \(M\) is a submonoid of the additive monoid of nonnegative rational numbers.https://www.zbmath.org/1453.130602021-02-27T13:50:00+00:00"Gipson, Ryan"https://www.zbmath.org/authors/?q=ai:gipson.ryan"Kulosman, Hamid"https://www.zbmath.org/authors/?q=ai:kulosman.hamidSummary: We investigate the atomicity and the AP property of the semigroup rings \(F[X; M]\), where \(F\) is a field, \(X\) is a variable and \(M\) is a submonoid of the additive monoid of nonnegative rational numbers. The main notion that we introduce for the purpose of the investigation is the notion of essential generators of \(M\).Iterated local cohomology groups and Lyubeznik numbers for determinantal rings.https://www.zbmath.org/1453.130532021-02-27T13:50:00+00:00"Lőrincz, András C."https://www.zbmath.org/authors/?q=ai:lorincz.andras-cristian"Raicu, Claudiu"https://www.zbmath.org/authors/?q=ai:raicu.claudiuThe authors give an explicit recipe for determining iterated local cohomology groups with support in ideals of
minors of a generic matrix in characteristic zero, expressing them as direct sums of indecomposable
D-modules. For nonsquare matrices these indecomposables are simple, but this is no longer true for
square matrices where the relevant indecomposables arise from the pole order filtration associated with
the determinant hypersurface. Theorem 1.1 determines the class in \(\Gamma_D\) of the local
cohomology groups of each \(D_p\), thus generalizing the main result of [\textit{C. Raicu} and \textit{J. Weyman}, Algebra Number Theory 8, No. 5, 1231--1257 (2014; Zbl 1303.13018)]
which addresses the case \(p = n\).
For nonsquare matrices, Theorem 1.1, together with the fact that \(\bmod_{\mathrm{GL}}(D_X )\) is semisimple, gives a description of Lyubeznik numbers. Specializing our results to a single iteration, they determine the Lyubeznik numbers for all generic determinantal rings and prove the vanishing of a range of local cohomology groups. Next, they use the quiver description of the
category \(\bmod_{\mathrm{GL}}(D_X )\) in conjunction with the vanishing results to provide an inductive proof of Theorem 1.6.
Reviewer: Ismael Akray (Soran)Equivariant higher Hochschild homology and topological field theories.https://www.zbmath.org/1453.570252021-02-27T13:50:00+00:00"Müller, Lukas"https://www.zbmath.org/authors/?q=ai:muller.lukas"Woike, Lukas"https://www.zbmath.org/authors/?q=ai:woike.lukasSummary: We present a version of higher Hochschild homology for spaces equipped with principal bundles for a structure group \(G\). As coefficients, we allow \(E_{\infty}\)-algebras with \(G\)-action. For this homology theory, we establish an equivariant version of excision and prove that it extends to an equivariant topological field theory with values in the \((\infty , 1)\)-category of cospans of \(E_{\infty}\)-algebras.Statement of retractions to: ``Fuzzification of ideals and filters in \(\Gamma\)-semigroups'' and ``On the socles of commutator invariant submodules of \(QTAG\)-modules''.https://www.zbmath.org/1453.200862021-02-27T13:50:00+00:00From the text: This is to notify our respectful reading public that the Editorial Board of the Armenian Journal of Mathematics has retracted the following articles from publication:
1. [\textit{A. Iampan}, Armen. J. Math. 4, No. 1, 44--48 (2012; Zbl 1281.20079)], this article substantially reproduced the content of an original article: [\textit{N. Kehayopulu} and \textit{M. Tsingelis}, Semigroup Forum 65, No. 1, 128--132 (2002; Zbl 1006.06008)].
2. [\textit{A. Hasan}, Armen. J. Math. 10, Paper No. 7, 11 p. (2018; Zbl 1403.16005)], this article reproduced the content of an original article by \textit{P. Danchev} and \textit{B. Goldsmith} [J. Group Theory 17, No. 5, 781--803 (2014; Zbl 1305.20062)] with only substitution ``QTAG-module'' instead of ``abelian group''.Annihilator conditions on modules over commutative rings.https://www.zbmath.org/1453.130282021-02-27T13:50:00+00:00"Anderson, D. D."https://www.zbmath.org/authors/?q=ai:anderson.daniel-d"Chun, Sangmin"https://www.zbmath.org/authors/?q=ai:chun.sangminWhen are multidegrees positive?https://www.zbmath.org/1453.140192021-02-27T13:50:00+00:00"Castillo, Federico"https://www.zbmath.org/authors/?q=ai:castillo.federico"Cid-Ruiz, Yairon"https://www.zbmath.org/authors/?q=ai:cid-ruiz.yairon"Li, Binglin"https://www.zbmath.org/authors/?q=ai:li.binglin"Montaño, Jonathan"https://www.zbmath.org/authors/?q=ai:montano.jonathan"Zhang, Naizhen"https://www.zbmath.org/authors/?q=ai:zhang.naizhenLet \(\mathbb{K}\) be an arbitrary field, \(\mathbb{P}= \mathbb{P}_{\mathbb{K}}^{m_1} \times \ldots \times \mathbb{P}_{\mathbb{K}}^{m_p}\) be a multiprojective space over \(\mathbb{K}\) and \(X \subseteq \mathbb{P}\) be a closed subscheme of \(\mathbb{P}.\) Let \(n= (n_1, \ldots, n_p) \in \mathbb{N}^{p}\) such that \(n_1+ \ldots+ n_p= \dim(X).\) They proved that the multidegree \(\deg_{\mathbb{P}}^{n}(X)\) is positive if and only if there is an irreducible component \(Y \subseteq X\) such that \( \dim(Y)= \dim(X)\) and for each \(J=\{j_1, \ldots, j_k\} \subseteq \{1, \ldots, p\}\) the inequality \(n_{j_1}+ \ldots+ n_{j_k} \leq \dim(\pi_{J}(Y)).\)
\textit{P. Brändén} and \textit{J. Huh} [Lorentzian polynomials'', Preprint, \url{arXiv:1902.03719}] showed that support of volume polynomial is M-convex. In the paper under review, the authors showed that \(MSupp_{\mathbb{P}}(X)= \{n \in \mathbb{N}^{p} | \deg_{\mathbb{P}}^{n}(X) > 0\} \) is discrete polymatroid. Also they defined another type of polymatroid called Chow polymatroids and proved that it is between linear polymatroids and Algebraic polymatroids.
From algebraic point of view, multidegree receive the name of mixed multiplicity. In the paper under review, the authors translated the above first result to the mixed multiplicities of a standard multigraded algebra over an Artinian local ring. This translation give a characterization for the positivity of mixed multiplicities of ideals.
Reviewer's rermark: There are some minor typos:
1- On page 28, there are additional parentheses in \(\prod_{J}(X)\).
2- On page 28, in the chain of \(V_i\), the second one is \(V_1\).
3- On page 31, the parentheses are missing on \(V(P;n)\).
Reviewer: Zahra Shahidi (Zanjan)Attached primes and annihilators of top local cohomology modules defined by a pair of ideals.https://www.zbmath.org/1453.130522021-02-27T13:50:00+00:00"Karimi, S."https://www.zbmath.org/authors/?q=ai:karimi.susan"Payrovi, Sh."https://www.zbmath.org/authors/?q=ai:payrovi.shiroyehAssume that $R$ is a complete Noetherian local ring and $M$ is a non-zero finitely generated $R$-module of dimension $n =\dim(M) > 1$. The authors prove that if $R$ is complete with respect to $m$-adic topology, then for any non-empty subset $T$ of Assh$(M)$, there exist ideals $I, J$ of $R$ such that $T=\mathrm{Att}_R(H^n_I(M)) = \mathrm{Att}_R(H^n_{I,J} (M))$. They define $T_R(I, J, M)$ to be the largest submodule of $M$ with cohomological dimension less than cohomological dimention of $M$. The authors prove that $T_R(I, J, M) = \Gamma_a (M) = \bigoplus_{cd(I,J, R/P_j)=c} N_j$, where $\bigoplus_{j=1}^n N_j=0$ denotes a reduced primary decomposition of the zero submodule of $M$, $N_j$ is a $P_j$ primary submodule of $M$ and $a = \bigoplus_c P_j$. Also the annihilator of $H^n_{I,J} (M)$ and $M/T_R(I, J, M)$ are equal, and some applications of this result are given.
Reviewer: Ismael Akray (Soran)A theory and an algorithm for computing sparse multivariate polynomial remainder sequence.https://www.zbmath.org/1453.130852021-02-27T13:50:00+00:00"Sasaki, Tateaki"https://www.zbmath.org/authors/?q=ai:sasaki.tateakiSummary: This paper presents an algorithm for computing the polynomial remainder sequence (PRS) and corresponding cofactor sequences of sparse multivariate polynomials over a number field \({\mathbb K}\). Most conventional algorithms for computing PRSs are based on the pseudo remainder (Prem), and the celebrated subresultant theory for the PRS has been constructed on the Prem. The Prem is uneconomical for computing PRSs of sparse polynomials. Hence, in this paper, the concept of sparse pseudo remainder (spsPrem) is defined. No subresultant-like theory has been developed so far for the PRS based on spsPrem. Therefore, we develop a matrix theory for spsPrem-based PRSs. The computational formula for PRS, regardless of whether it is based on Prem or spsPrem, causes a considerable intermediate expression growth. Hence, we next propose a technique to suppress the expression growth largely. The technique utilizes the power-series arithmetic but no Hensel lifting. Simple experiments show that our technique suppresses the intermediate expression growth fairly well, if the sub-variable ordering is set suitably.
For the entire collection see [Zbl 1396.68014].Factoring multivariate polynomials with many factors and huge coefficients.https://www.zbmath.org/1453.130722021-02-27T13:50:00+00:00"Monagan, Michael"https://www.zbmath.org/authors/?q=ai:monagan.michael-b"Tuncer, Baris"https://www.zbmath.org/authors/?q=ai:tuncer.barisSummary: The standard approach to factor a multivariate polynomial in \(\mathbb{Z}[x_1,x_2,\ldots,x_n]\) is to factor a univariate image in \(\mathbb{Z}[x_1]\) then recover the multivariate factors from their images using a process known as multivariate Hensel lifting. For the case when the factors are expected to be sparse, at CASC 2016, we introduced a new approach which uses sparse polynomial interpolation to solve the multivariate polynomial diophantine equations that arise inside Hensel lifting.
In this work we extend our previous work to the case when the number of factors to be computed is more than 2. Secondly, for the case where the integer coefficients of the factors are large we develop an efficient \(p\)-adic method. We will argue that the probabilistic sparse interpolation method introduced by us provides new options to speed up the factorization for these two cases. Finally we present some experimental data comparing our new methods with previous methods.
For the entire collection see [Zbl 1396.68014].Effective localization using double ideal quotient and its implementation.https://www.zbmath.org/1453.130122021-02-27T13:50:00+00:00"Ishihara, Yuki"https://www.zbmath.org/authors/?q=ai:ishihara.yuki"Yokoyama, Kazuhiro"https://www.zbmath.org/authors/?q=ai:yokoyama.kazuhiroSummary: In this paper, we propose a new method for localization of polynomial ideal, which we call ``local primary algorithm''. For an ideal \(I\) and a prime ideal \(P\), our method computes a \(P\)-primary component of \(I\) after checking if \(P\) is associated with \(I\) by using double ideal quotient \((I:(I:P))\) and its variants which give us a lot of information about localization of \(I\).
For the entire collection see [Zbl 1396.68014].Totally reflexive modules over rings that are close to Gorenstein.https://www.zbmath.org/1453.130032021-02-27T13:50:00+00:00"Kustin, Andrew R."https://www.zbmath.org/authors/?q=ai:kustin.andrew-r"Vraciu, Adela"https://www.zbmath.org/authors/?q=ai:vraciu.adela-nOver a commutative Noetherian ring, \textit{M. Auslander} and \textit{M. Bridger} [Stable module theory. Providence, RI: American Mathematical Society (AMS) (1969; Zbl 0204.36402)] extended the notion of finitely generated projective modules to totally reflexive modules, i.e. modules occurring as the cokernels of differentials in exact chain complexes of finitely generated free modules whose dual complexes are also exact. There has been a growing interest in identifying the local rings over which every totally reflexive module is projective or equivalently free, i.e. the so-called ``G-regular rings'' coined by Takahashi. Regular local rings and Golod local rings that are not hypersurfaces are among the examples of G-regular rings. However, a singular Gorenstein local ring is never G-regular. In the paper under review, the authors prove that any non-Gorenstein quotient of small colength of a deeply embedded equicharacteristic Artinian Gorenstein local ring is G-regular. Their investigation relies on the fact that an Artinian local ring is G-regular if there exists a projective-test module that happens to be a direct summand of a syzygy of the canonical module of the ring. Overall, this is an interesting and readable paper.
Reviewer: Hossein Faridian (Clemson)The geometry of gaussoids.https://www.zbmath.org/1453.130762021-02-27T13:50:00+00:00"Boege, Tobias"https://www.zbmath.org/authors/?q=ai:boege.tobias"D'Alì, Alessio"https://www.zbmath.org/authors/?q=ai:dali.alessio"Kahle, Thomas"https://www.zbmath.org/authors/?q=ai:kahle.thomas"Sturmfels, Bernd"https://www.zbmath.org/authors/?q=ai:sturmfels.berndThis work arises from trying to answer Problem 4 presented in [\textit{B. Sturmfels}, IMA Vol. Math. Appl. 149, 351--363 (2009; Zbl 1158.13300)]: ``General problem: Study the geometry of conditional independence models for multivariate Gaussian random variables.'' Thus, this survey develops the geometric theory of gaussoids, based on the Lagrangian Grassmannian and its symmetries. A gaussoid is a combinatorial structure that encodes independence in probability and statistics, just like matroids encode independence in linear algebra. Throughout the 30 pages, the authors introduce and classify oriented gaussoids, connect valuated gaussoids to tropical geometry, addresses the realizability problem for gaussoids and oriented gaussoids and so on. The reader can find additional materials on the web \url{www.gaussoids.de}.
Reviewer: Gema Maria Diaz Toca (Murcia)Existence of almost Cohen-Macaulay algebras implies the existence of big Cohen-Macaulay algebras.https://www.zbmath.org/1453.130322021-02-27T13:50:00+00:00"Bhattacharyya, Rajsekhar"https://www.zbmath.org/authors/?q=ai:bhattacharyya.rajsekharSummary: In [\textit{M. Asgharzadeh} and the author, J. Algebra Appl. 11, No. 4, Article ID 1250075, 10 p. (2012; Zbl 1255.13008)], the dagger closure is extended over finitely generated modules over Noetherian local domain \((R, \mathfrak{m})\) and it is proved to be a Dietz closure. In this short note we show that it also satisfies the `Algebra axiom' of [\textit{R. R. G.}, J. Pure Appl. Algebra 222, No. 7, 1878--1897 (2018; Zbl 06846335)] and this leads to the following result of this paper: For a complete Noetherian local domain, if it is contained in an almost Cohen-Macaulay domain, then there exists a balanced big Cohen-Macaulay algebra over it.Algorithms for commutative algebras over the rational numbers.https://www.zbmath.org/1453.130582021-02-27T13:50:00+00:00"Lenstra, H. W. jun."https://www.zbmath.org/authors/?q=ai:lenstra.hendrik-w-jun"Silverberg, A."https://www.zbmath.org/authors/?q=ai:silverberg.aliceSummary: The algebras considered in this paper are commutative rings of which the additive group is a finite-dimensional vector space over the field of rational numbers. We present deterministic polynomial-time algorithms that, given such an algebra, determine its nilradical, all of its prime ideals, as well as the corresponding localizations and residue class fields, its largest separable subalgebra, and its primitive idempotents. We also solve the discrete logarithm problem in the multiplicative group of the algebra. While deterministic polynomial-time algorithms were known earlier, our approach is different from previous ones. One of our tools is a primitive element algorithm; it decides whether the algebra has a primitive element and, if so, finds one, all in polynomial time. A methodological novelty is the use of derivations to replace a Hensel-Newton iteration. It leads to an explicit formula for lifting idempotents against nilpotents that is valid in any commutative ring.Computation of Pommaret bases using syzygies.https://www.zbmath.org/1453.130752021-02-27T13:50:00+00:00"Binaei, Bentolhoda"https://www.zbmath.org/authors/?q=ai:binaei.bentolhoda"Hashemi, Amir"https://www.zbmath.org/authors/?q=ai:hashemi.amir"Seiler, Werner M."https://www.zbmath.org/authors/?q=ai:seiler.werner-mSummary: We investigate the application of syzygies for efficiently computing (finite) Pommaret bases. For this purpose, we first describe a non-trivial variant of Gerdt's algorithm [\textit{V. P. Gerdt}, in: Computational commutative and non-commutative algebraic geometry. Proceedings of the NATO Advanced Research Workshop, Chisinau, Republic of Moldova, 2004. Amsterdam: IOS Press, 199--225 (2005; Zbl 1104.13012)] to construct an involutive basis for the input ideal as well as an involutive basis for the syzygy module of the output basis. Then we apply this new algorithm in the context of Seiler's method to transform a given ideal into quasi stable position to ensure the existence of a finite Pommaret basis [\textit{W. M. Seiler},
Appl. Algebra Eng. Commun. Comput. 20, No. 3--4, 261--338 (2009; Zbl 1175.13011)]. This new approach allows us to avoid superfluous reductions in the iterative computation of Janet bases required by this method. We conclude the paper by proposing an involutive variant of the signature based algorithm of \textit{S. Gao} et al. [Math. Comput. 85, No. 297, 449--465 (2016; Zbl 1331.13018)] to compute simultaneously a Gröbner basis for a given ideal and for the syzygy module of the input basis. All the presented algorithms have been implemented in Maple and their performance is evaluated via a set of benchmark ideals.
For the entire collection see [Zbl 1396.68014].Undirected zero-divisor graphs and unique product monoid rings.https://www.zbmath.org/1453.160402021-02-27T13:50:00+00:00"Hashemi, Ebrahim"https://www.zbmath.org/authors/?q=ai:hashemi.ebrahim"Alhevaz, Abdollah"https://www.zbmath.org/authors/?q=ai:alhevaz.abdollahLet \(R\) be an associative ring with identity and \(Z(R)\) be its set of non-zero zero-divisors. The zero-divisor graph \(\Gamma(R)\) of \(R,\) is the simple undirected graph with vertex set \(Z^*(R)=Z(R)\setminus \{0\}\) where two distinct vertices \(x\) and \(y\) are adjacent if and only if \(xy=0\) or \(yx = 0.\) In fact, this is a generalization of the zero divisor graph of a commutative ring \(R\) introduced by \textit{D. F. Anderson} and \textit{P. S. Livingston} [J. Algebra 217, No. 2, 434--447 (1999; Zbl 0941.05062)] and studied by several authors for the past three decades. The distance between vertices \(a\) and \(b\) is the length of the shortest path connecting them, and the diameter of the graph, diam(\(\Gamma(R)\)), is the supremum of these distances between all pairs of distinct vertices in \(\Gamma(R).\)
In this paper, the authors first prove some results about \(\Gamma(R)\), where \(R\) is a semi-commutative ring or a reversible ring or a unique product monoid \(M.\) Using those results, it is proved that \(0\leq \mathrm{diam}(\Gamma(R)) \leq \mathrm{diam}(\Gamma(R[M]))\leq 3.\) Having obtained upper and lower bounds for the diameter of \(\Gamma(R)\) and \(\Gamma(R[m]),\) they characterize all the possibilities for the pair \(\mathrm{diam}(\Gamma(R))\) and \(\mathrm{diam}(\Gamma(R[M])),\) strictly in terms of the properties of the underlying ring \(R,\) where \(R\) is a reversible ring and \(M\) is a unique product monoid. At the last part of the paper, the authors provide an example showing the necessity of the assumptions used in the statement of the result.
Reviewer: T. Tamizh Chelvam (Tirunelveli)A characterization of the Arf property for quadratic quotients of the Rees algebra.https://www.zbmath.org/1453.200782021-02-27T13:50:00+00:00"Borzì, Alessio"https://www.zbmath.org/authors/?q=ai:borzi.alessioThe paper is focused in the study of Arf numerical semigroups and Arf rings and a characterization of this property is given for the numerical duplication of numerical semigroups and for quotiens of Rees algebras.
In the first part of the paper, some basic definitions about this topic are shown and the author studies how the property of being Arf and the numerical duplication are related. If \(S\) is a numerical semigroup, \(E\) is an semigroup ideal, \(m\) is an element of \(S\) and \(S\Join^m E\) is the numerical duplication with respect to \(E\) and \(m\), then some conditions are given such that \(\text{Arf}(S)\text{Join}^m\bar{E}=\text{Arf}(S\text{Join}^m E)\) holds where \(\bar{E}\) and \(\text{Arf}(S)\) denote the closure of \(E\) in \(S\) and the Arf clausure, respectively.
Finally, let \(R\) be a Noetherian, analytically irreducible, residually rational, one-dimensional, local domain with \(\text{char}(R)\neq 2\), \(I\) an ideal of \(R\) and \(t\) an indeterminate, the Rees algebra associated with \(R\) and \(I\) is \(R[It]\). Let \(v\) be the valuation on \(Q(R)\), \(b\in R\) such that \(v(b)\) is odd and \(\mathcal{R}=\frac{R[It]}{(t^2-b)\cap R[It]}\), then the author shows how \(R\) and \(\mathcal{R}\) are both Arf under some hypothesis.
Reviewer: Daniel Marín Aragon (Cádiz)Existence of Euclidean ideal classes beyond certain rank.https://www.zbmath.org/1453.111332021-02-27T13:50:00+00:00"Sivaraman, Jyothsnaa"https://www.zbmath.org/authors/?q=ai:sivaraman.jyothsnaaIt has been conjectured by \textit{H. W. Lenstra} [Astérisque 61, 121--131 (1979; Zbl 0401.12005)] that the ring of integers of an algebraic number field \(K\) having infinitely many units possesses an Euclidean ideal if and only its class-group is cyclic. Lenstra showed also that this conjecture is a consequence of \textit{GRH\/}. Later \textit{H. Graves} and \textit{M. Ram Murty} [Proc. Am. Math. Soc. 141, 2979--2990 (2013; Zbl 1329.11115)] established this conjecture for all fields having abelian Hilbert class fields \(H(K)\) and unit rank \(\ge4\). The author provides a new proof of a special case of this result with unspecified lower limit for the unit rank and assuming additionally that \(H(K)\) is a subfield of a cyclotomic field with cyclic Galois group. The main tool of the proof is a variant of Brun's sieve in the form given by \textit{Y.-F. Bilu} et al. [Compos. Math. 154, 2441--2461 (2018; Zbl 1444.11071)], whereas Gupta and Murty utilized the linear sieve.
Reviewer: Władysław Narkiewicz (Wrocław)Lyubeznik and Betti numbers for homogeneous ideals.https://www.zbmath.org/1453.130542021-02-27T13:50:00+00:00"Nadi, Parvaneh"https://www.zbmath.org/authors/?q=ai:nadi.parvaneh"Rahmati, Farhad"https://www.zbmath.org/authors/?q=ai:rahmati.farhadAuthors' abstract: Let \(R = K[x_1, \ldots, x_n]\) be the standard graded polynomial ring over a field \(K.\) We'll study the Lyubeznik numbers of two homogeneous ideals of \(R\) whose initial ideals are linked for some monomial order over \(R.\) We'll also investigate some relations between Lyubeznik and Betti numbers of homogeneous ideals.
Reviewer: Kriti Goel (Mumbai)Polyhedral parametrizations of canonical bases \& cluster duality.https://www.zbmath.org/1453.130652021-02-27T13:50:00+00:00"Genz, Volker"https://www.zbmath.org/authors/?q=ai:genz.volker"Koshevoy, Gleb"https://www.zbmath.org/authors/?q=ai:koshevoy.gleb-a"Schumann, Bea"https://www.zbmath.org/authors/?q=ai:schumann.beaSummary: We establish the relation of Berenstein-Kazhdan's decoration function and Gross-Hacking-Keel-Kontsevich's potential on the open double Bruhat cell in the base affine space \(\text{G} / \mathcal{N}\) of a simple, simply connected, simply laced algebraic group \(G\). As a byproduct we derive explicit identifications of polyhedral parametrization of canonical bases of the ring of regular functions on \(\text{G} / \mathcal{N}\) arising from the tropicalizations of the potential and decoration function with the classical string and Lusztig parametrizations. In the appendix we construct maximal green sequences for the open double Bruhat cell in \(\text{G} / \mathcal{N}\) which is a crucial assumption for Gross-Hacking-Keel-Kontsevich's construction.Regularity of \(\mathfrak{S}_n\)-invariant monomial ideals.https://www.zbmath.org/1453.130552021-02-27T13:50:00+00:00"Raicu, Claudiu"https://www.zbmath.org/authors/?q=ai:raicu.claudiuLet \(S\) be the polynomial ring in \(n\) variables and \(\mathfrak{S}_n\) the symmetric group acting on \(S\) by permuting the variables. The present paper considers \(\mathfrak{S}_n\)-invariant monomial ideals \(I\subseteq S\).
The main theorem in the paper gives a recipe for the explicit computation of the modules \(\operatorname{Ext}^j_S(S/I,S)\) for all \(j\geq 0\) (Theorem 3.1). This result allows, among other consequences, to describe the projective dimension and regularity of \(I\) (Formula (1.3) and Section 3.1).
The paper also classfies the \(\mathfrak{S}_n\)-invariant monomial ideals that have a linear free resolution (Theorem 4.4) and characterizes those which are Cohen-Macaulay (Section 3.2).
Finally, the author analyzes the asymptotic behaviour of regularity in two different settings: considering powers of a fixed ideal \(I\) (Theorem 5.1) and varying the dimension of the ambient polynomial ring and examining the invariant monomial ideals induced by a given ideal \(I\) (Theorem 6.1).
Reviewer: Eduardo Saenz-de-Cabezon (Logroño)On a polytime factorization algorithm for multilinear polynomials over \(\mathbb{F}_2\).https://www.zbmath.org/1453.120012021-02-27T13:50:00+00:00"Emelyanov, Pavel"https://www.zbmath.org/authors/?q=ai:emelyanov.pavel-g"Ponomaryov, Denis"https://www.zbmath.org/authors/?q=ai:ponomaryov.denis-kSummary: In 2010, \textit{A. Shpilka} and \textit{I. Volkovich} [ICALP 2010, Lect. Notes Comput. Sci. 6198, 408--419 (2010; Zbl 1288.68252)] established a prominent result on the equivalence of polynomial factorization and identity testing. It follows from their result that a multilinear polynomial over the finite field of order 2 can be factored in time cubic in the size of the polynomial given as a string. Later, we have rediscovered this result and provided a simple factorization algorithm based on computations over derivatives of multilinear polynomials. The algorithm has been applied to solve problems of compact representation of various combinatorial structures, including Boolean functions and relational data tables. In this paper, we describe an improvement of this factorization algorithm and report on preliminary experimental analysis.
For the entire collection see [Zbl 1396.68014].Basis criteria for generalized spline modules via determinant.https://www.zbmath.org/1453.051112021-02-27T13:50:00+00:00"Altınok, Selma"https://www.zbmath.org/authors/?q=ai:altinok.selma"Sarıoğlan, Samet"https://www.zbmath.org/authors/?q=ai:sarioglan.sametSummary: Given a graph whose edges are labeled by ideals of a commutative ring \(R\) with identity, a generalized spline is a vertex labeling by the elements of \(R\) such that the difference of the labels on adjacent vertices lies in the ideal associated to the edge. The set of generalized splines has a ring and an \(R\)-module structure. We study the module structure of generalized splines where the base ring is a greatest common divisor domain. We give basis criteria for generalized splines on cycles, diamond graphs and trees by using determinantal techniques. In the last section of the paper, we define a graded module structure for generalized splines and give some applications of the basis criteria for cycles, diamond graphs and trees.Constructing Fano 3-folds from cluster varieties of rank 2.https://www.zbmath.org/1453.130642021-02-27T13:50:00+00:00"Coughlan, Stephen"https://www.zbmath.org/authors/?q=ai:coughlan.stephen"Ducat, Tom"https://www.zbmath.org/authors/?q=ai:ducat.tomThe authors start by giving an introduction to cluster varieties and Gorenstein formats. Then they focus on \(C_2\) and \(G_2\) rank 2 cluster formats in more detail and explain some ways of constructing them. This introduction is very welcome for non-experts, as it explains the basic concepts and gives several enlightening examples.
Afterwards they apply these
formats to construct Fano 3-folds and give many examples. They finish explaining the computer algorithm that they use to make theirs classification.
The main application
is to construct hundreds of families of Fano 3-folds in codimensions 4 and 5. In
particular, for Fano 3-folds in codimension 4 they construct at least one family for 187 of
the 206 possible Hilbert polynomials contained in the Graded Ring Database.
Reviewer: Rick Rischter (Itajubá)An alternating matrix and a vector, with application to Aluffi algebras.https://www.zbmath.org/1453.130682021-02-27T13:50:00+00:00"Kustin, Andrew R."https://www.zbmath.org/authors/?q=ai:kustin.andrew-rSummary: Let \(\mathbf{X}\) be a generic alternating matrix, t be a generic row vector, and \(J\) be the ideal \(\operatorname{Pf}_4(\mathbf{X}) + I_1(\mathbf{tX})\). We prove that \(J\) is a perfect Gorenstein ideal of grade equal to the grade of \(\operatorname{Pf}_4(\mathbf{X})\) plus two. This result is used by Ramos and Simis in their calculation of the Aluffi algebra of the module of derivations of the homogeneous coordinate ring of a smooth projective hypersurface. We also prove that \(J\) defines a domain, or a normal ring, or a unique factorization domain if and only if the base ring has the same property. The main object of study in the present paper is the module \(\mathcal{N}\) which is equal to the column space of \(\mathbf{X}\), calculated mod \(\operatorname{Pf}_4(\mathbf{X})\). The module \(\mathcal{N}\) is a self-dual maximal Cohen-Macaulay module of rank two; furthermore, \(J\) is a Bourbaki ideal for \(\mathcal{N}\). The ideals which define the homogeneous coordinate rings of the Plücker embeddings of the Schubert subvarieties of the Grassmannian of planes are used in the study of the module \(\mathcal{N}\).Kummer theories for algebraic tori and normal basis problem.https://www.zbmath.org/1453.130252021-02-27T13:50:00+00:00"Suwa, Noriyuki"https://www.zbmath.org/authors/?q=ai:suwa.noriyukiSummary: We discuss the inverse Galois problem with normal basis, concerning Kummer theories for algebraic tori, in the framework of group schemes. The unit group scheme of a group algebra plays an important role in this article, as was pointed out by \textit{J.-P. Serre} [Groupes algébriques et corps de classes. Paris: Hermann \& Cie (1959; Zbl 0097.35604)]. We develop our argument not only over a field but also over a ring, considering integral models of Kummer theories for algebraic tori.Toric degenerations of Grassmannians and Schubert varieties from matching field tableaux.https://www.zbmath.org/1453.141192021-02-27T13:50:00+00:00"Clarke, Oliver"https://www.zbmath.org/authors/?q=ai:clarke.oliver"Mohammadi, Fatemeh"https://www.zbmath.org/authors/?q=ai:mohammadi.fatemehSummary: We study Gröbner degenerations of Grassmannians and the Schubert varieties inside them. We provide a family of binomial ideals whose combinatorics is governed by matching field tableaux in the sense of \textit{B. Sturmfels} and \textit{A. V. Zelevinsky} [Adv. Math. 98, No. 1, 65--112 (1993; Zbl 0776.13009)]. We prove that these ideals are all quadratically generated and they yield a SAGBI basis of the Plücker algebra. This leads to a new family of toric degenerations of Grassmannians. Moreover, we apply our results to construct a family of Gröbner degenerations of Schubert varieties inside Grassmannians. We provide a complete characterization of toric ideals among these degenerations in terms of the combinatorics of matching fields, permutations and semi-standard tableaux.Symmetric decomposition of the associated graded algebra of an Artinian Gorenstein algebra.https://www.zbmath.org/1453.130672021-02-27T13:50:00+00:00"Iarrobino, Anthony"https://www.zbmath.org/authors/?q=ai:iarrobino.anthony-a"Macias Marques, Pedro"https://www.zbmath.org/authors/?q=ai:marques.pedro-maciasLet \(k\) be an arbitrary field, \(R = k\{x_1,\dots x_r \}\) the completed local ring in \(r\) variables, and let \(\mathfrak D = k_{DP}[X_1,\dots,X_r]\) the divided power algebra in \(X_1,\dots,X_r\). We take \(R\) as acting on \(\mathfrak D\) by contraction. Macaulay showed that giving an ideal \(I\) of \(R\) defining an artinian quotient \(A=R/I\) of length \(n\) is equialent to giving a length \(n\) \(R\)-submodule \(\hat A\) of \(\mathfrak D\). \(\hat A\) is called the \textit{Macaulay inverse system \(\hat A = I^\perp\) of \(I\)}. \(\hat A\) is \textit{Gorenstein} if \(\hat A\) has a single generator \(f_A\). From now on we assume \(\hat A\) is Gorenstein. Let \(\mathfrak m_A\) denote the maximum ideal of \(A\); then the \textit{socle degree} \(j_A\) of \(A\) is \(\max \{ i \ | \ \mathfrak m_A^i \neq 0\}\). The degree of \(f_A\) is equal to \(j_A\), and \(f_A\) generates the cylic \(A\)-module \(\hat A\). The first author showed, many years ago, that the associated graded algebra \(A^* = Gr_{m_A} (A)\) of \(A\) has a canonical stratification by ideals \(C(a) = C_A(a)\) whose successive quotients \(Q(a) = Q_A(a) \cong C(a) / C(a+1)\) are reflexive \(A^*\) modules; these \(Q(a)\) are the \textit{symmetric subquotients} of \(A^*\). The Hilbert function \(H(A)\) may be written as a sum of symmetric sequences \(H_A(a) = H(Q_A(a))\); these sequences are the \textit{symmetric components} of the Hilbert function, and the \textit{symmetric decomposition} \(\mathcal D(A)\) of \(H(A)\) is the sequence \(\mathcal D(A) = (H_A(0), H_A(1), \dots, H_A(j_A))\).
This paper studies this symmetric subquotient decomposition of \(A^*\). The authors study which sequences may occur as these summands, and construct examples of artinian Gorenstein algebras for which \(H_A(a)\) can have interior zeros, for instance of the form \(H_A(a) = (0,s,0,\dots,0,s,0)\). They determine which sequences \(H_A(a)\) can be non-zero when the dual generator is linear in a subset of the variables. They also study so-called \textit{exotic summands} of the Macaulay dual generator, and apply this to Gorenstein algebras that are connected sums.
Reviewer: Juan C. Migliore (Notre Dame)Leaps of modules of integrable derivations in the sense of Hasse-Schmidt.https://www.zbmath.org/1453.130702021-02-27T13:50:00+00:00"Tirado Hernández, María de la Paz"https://www.zbmath.org/authors/?q=ai:tirado-hernandez.maria-de-la-pazSummary: Let \(k\) be a commutative ring of characteristic \(p > 0\). We prove that leaps of chain formed by modules of integrable derivations in the sense of Hasse-Schmidt of a \(k\)-algebra only occur at powers of \(p\).On the exponents of free and nearly free projective plane curves.https://www.zbmath.org/1453.140862021-02-27T13:50:00+00:00"Dimca, Alexandru"https://www.zbmath.org/authors/?q=ai:dimca.alexandru"Sticlaru, Gabriel"https://www.zbmath.org/authors/?q=ai:sticlaru.gabrielSummary: We show that all the possible pairs of integers occur as exponents for free or nearly free irreducible plane curves and line arrangements, by producing only two types of simple families of examples. The topology of the complements of these curves and line arrangements is also discussed, and many of them are shown not to be \(K(\pi ,1)\) spaces.A generalization of the prime radical of ideals in commutative rings.https://www.zbmath.org/1453.130102021-02-27T13:50:00+00:00"Harehdashti, Javad Bagheri"https://www.zbmath.org/authors/?q=ai:harehdashti.javad-bagheri"Moghimi, Hosein Fazaeli"https://www.zbmath.org/authors/?q=ai:moghimi.hosein-fazaeliSummary: Let \(R\) be a commutative ring with identity, and \(\phi : \mathcal{I} (R) \rightarrow \mathcal{I} (R) \cup \{\varnothing\}\) be a function where \(\mathcal{I} (R)\) is the set of all ideals of \(R\). Following [\textit{D. D. Anderson} and \textit{M. Bataineh}, Commun. Algebra 36, No. 2, 686--696 (2008; Zbl 1140.13005)], a proper ideal \(P\) of \(R\) is called a \(\phi\)-prime ideal if \(x, y \in R\) with \(xy \in P - \phi(P)\)) implies \(x \in P\) or \(y \in P\). For an ideal \(I\) of \(R\), we define the \(\phi\)-radical \(\sqrt [\phi] {I}\) to be the intersection of all \(\phi\)-prime ideals of \(R\) containing \(I\), and show that this notion inherits most of the essential properties of the usual notion of radical of an ideal. We also investigate when the set of all \(\phi\)-prime ideals of \(R\), denoted \(\mathrm{Spec}_\phi (R)\), has a Zariski topology analogous to that of the prime spectrum \(\mathrm{Spec}(R)\), and show that this topological space is Noetherian if and only if \(\phi\)-radical ideals of \(R\) satisfy the ascending chain condition.On the sequential polynomial type of modules.https://www.zbmath.org/1453.130562021-02-27T13:50:00+00:00"Goto, Shiro"https://www.zbmath.org/authors/?q=ai:goto.shiro"Nhan, Le Thanh"https://www.zbmath.org/authors/?q=ai:le-thanh-nhan.Summary: Let \(M\) be a finitely generated module over a Noetherian local ring \(R\). The sequential polynomial type \(\mathrm{sp}(M)\) of \(M\) was recently introduced by Nhan, Dung and Chau, which measures how far the module \(M\) is from the class of sequentially Cohen-Macaulay modules. The present paper purposes to give a parametric characterization for \(M\) to have \(\mathrm{sp}(M)\leq s\), where \(s\geq-1\) is an integer. We also study the sequential polynomial type of certain specific rings and modules. As an application, we give an inequality between \(\mathrm{sp}(S)\) and \(\mathrm{sp}(S^G)\), where \(S\) is a Noetherian local ring and \(G\) is a finite subgroup of \(\mathrm{Aut}S\) such that the order of \(G\) is invertible in \(S\).Frobenius powers of some monomial ideals.https://www.zbmath.org/1453.130202021-02-27T13:50:00+00:00"Hernández, Daniel J."https://www.zbmath.org/authors/?q=ai:hernandez.daniel-j"Teixeira, Pedro"https://www.zbmath.org/authors/?q=ai:teixeira.pedro"Witt, Emily E."https://www.zbmath.org/authors/?q=ai:witt.emily-eLet \(R=k[x_1,\ldots,x_n]\) be a polynomial ring of prime characteristic \(p\) with maximal ideal \(\mathfrak{m}=(x_1, \ldots, x_n)\). For any ideal \(\mathfrak{a}\) in \(R\) and \(t\) a non-negative real number, the authors define the generalized Frobenius power \(\mathfrak{a}^{[t]}\) which agrees with the Frobenius powers when \(t=p^e\) with \(e \in \mathbb{Z}\). They first define the Frobenius power for positive integers \(m=m_0+m_1p+\cdots m_rp^r\) to be \[\mathfrak{a}^{[m]}:=\mathfrak{a}^{m_0}(\mathfrak{a}^{m_1})^{[p]}\cdots (\mathfrak{a}^{m_r})^{[p^r]},\] then for rational numbers of the form \(m/p^e\) with \(\text{gcd}(m,p)=1\): \(\mathfrak{a}^{[m/p^e]}:=(\mathfrak{a}^{[m]})^{[1/p^e]}\). For a general non-negative \(t\), they define \(\mathfrak{a}^{[t]}=\bigcup_{k=1}^{\infty} \mathfrak{a}^{[t_k]}\) where \(t_k=m_k/p^{r_k}\) is a monotonically non-increasing sequence of positive rational numbers converging to \(t\). A non-negative real number \(\lambda\) is a critical exponent of an ideal \(\mathfrak{a}\) if \(\mathfrak{a}^{[\lambda]} \subsetneq \mathfrak{a}^{[\lambda-\epsilon]}\) for \(0 < \epsilon\leq \lambda\). The critical exponent of \(\mathfrak{a}\) with respect to \(\mathfrak{b}\) is \[\text{crit}(\mathfrak{a},\mathfrak{b})=\text{sup}\{t>0 \mid \mathfrak{a}^{[t]} \nsubseteq \mathfrak{b}\}= \text{min}\{t>0 \mid \mathfrak{a}^{[t]} \subseteq \mathfrak{b}\}.\] Some of the main goals of this paper are determining the critical exponents in the interval \([0,1]\) of powers of \(\mathfrak{m}\) and diagonal ideals \((x_1^{a_1}, \ldots, x_n^{a_n})\). As a preliminary step, they show that for every \(\mathfrak{m}\)-primary monomial ideal \(\mathfrak{a}\) the critical exponents are of the form \(\text{crit}(\mathfrak{a},(x_1^{u_1}, \ldots x_n^{u_n}))\) for some \((u_1, \ldots, u_n) \in \mathbb{N}^n\). They further show that the critical exponents \(\lambda \in [0,1]\) of \(\mathfrak{m}^d\) are of the form \[\lambda=\displaystyle\frac{k}{d}-\displaystyle\frac{[kp^s\% d]}{dp^s}\] where \([kp^e \% d] \) is the remainder of \(kp^e\) modulo \(d\) and \(s\) is the infinmum of the set of all \(e \geq 1\) where the remainders are less than \(n\). For each critical exponent \(\lambda\) as above \((\mathfrak{m}^d)^{[\lambda]}=\mathfrak{m}^{k-n+1}\). Using similar techniques they give expressions for the critical exponents of diagonal ideals. The many explicit examples given help to visualize their results for the critical exponents and the Frobenius powers for \(t \in [0,1]\).
They conclude with discussing the relationship between \(F\)-jumping numbers of a polynomial of the form \(f=\alpha_1x_1^{a_1}+\cdots \alpha_n x_n^{a_n} \in \mathfrak{m}^d\) with the critical exponents of monomial ideals. In particular, they show that if the \(\alpha_i\) are algebraically independent then \(\tau(f^t)=(\mathfrak{m}^d)^{[t]}\) for \(t \in (0,1)\). Whereas if the \(\alpha_i\) are nonzero, not necessarily algebraically independent and \(p\) doesn't divide \(a_i\) then \(\tau(f^t)=(x_1^{a_1}, \ldots,x_n^{a_n})^{[t]}\) for \(t \in (0,1)\).
Reviewer: Janet Vassilev (Albuquerque)Dimensions of multi-fan duality algebras.https://www.zbmath.org/1453.130662021-02-27T13:50:00+00:00"Ayzenberg, Anton"https://www.zbmath.org/authors/?q=ai:ayzenberg.anton-aThe paper under review considers the dimension vector of the graded Poincare duality algebra associated to a complete simplicial multi-fan.
Therefore, one starts with a multi-fan \(\Delta\) in an oriented space \(V\cong \mathbb{R}^n\), whose cones are simplicial and may overlap. Then one considers the pair \((\omega, \lambda)\), where
\[
\omega=\sum_{I\subset [m],|I|=n}\omega(I)I\in Z_{n-1}(\Delta_{[m]}^{(n-1)};\mathbb{R})
\]
is a simplicial cycle on \(m\) vertices, and \(\lambda:[m]\to V\) is a function such that \(\{\lambda(i)\mid i\in I\}\) is a basis of \(V\) if \(|I|=n\) and \(\omega(I)\ne 0\).
With respect to this \(\Delta\), one also need to consider the simple multi-polytope \(P=(\Delta,\{H_1,\dots,H_m\})\) with the support parameters \(c_1,\dots,c_m\in \mathbb{R}\). Here, \(H_i=\{x\in V^*\mid \langle x,\lambda(i)\rangle =c_i\}\) is a hyperplane in \(V^*\) perpendicular to \(\lambda(i)\in V\). This multi-polytope has a well-defined volume depending on the support parameters, which leads to the volume polynomial \(V_\Delta \in \mathbb{R}[c_1,\dots,c_m]\).
Now the graded Poincare duality algebra under consideration is simply \(\mathcal{A}^*(\Delta):=\mathcal{D}/ \{D\in \mathcal{D}\mid D V_\Delta=0\}\). Here, \(\mathcal{D}=\mathbb{R}[\partial_1,\dots,\partial_m]\), where \(\partial_i\) is the partial derivative operator \(\partial/\partial c_i\).
The paper under review first considers the question: whether the dimensions of graded components of \(\mathcal{A}^*(\Delta)\) depend only on \(\omega\), but not on \(\lambda\)? As found out by this paper, the dimensions do not depend on the values of \(\lambda\) at the non-singular vertices, but may depend crucially on the values of \(\lambda\) at the singular vertices.
The next question is then: what can be said when \(\omega\) is the fundamental cycle of an \((n-1)\)-dimensional oriented simplicial pseudomanifold \(K\), i.e., when the multi-fan is supported on \(K\)? Let \(r(K)\) be the number of distinct dimension vectors of multi-fans on \(K\). Then, this paper shows that \(r(K)=1\) when \(K\) is a homology manifold, or a \(3\)-dimensional pseudomanifold with isolated singularities. But \(r(K)\) might be nontrivial in general.
Another class of examples studied in this paper is by considering a link (namely a collection of knots in \(S^3\)) \(\ell:\bigsqcup_\alpha S_\alpha^1 \hookrightarrow S^3\) and then collapsing each of its components to a point. The examples computed by the author lead to the interesting question: is it true that in this case \(r(K)=1\) if and only if the components in \(\ell\) are pairwise unlinked?
Reviewer: Yi-Huang Shen (Hefei)Duality and symmetry of complexity over complete intersections via exterior homology.https://www.zbmath.org/1453.130472021-02-27T13:50:00+00:00"Liu, Jian"https://www.zbmath.org/authors/?q=ai:liu.jian.4|liu.jian.3|liu.jian.2|liu.jian.5|liu.jian|liu.jian.1|liu.jian.6"Pollitz, Josh"https://www.zbmath.org/authors/?q=ai:pollitz.joshLet \(R\) be a locally complete intersection ring. The authors prove that every thick subcategory \(\mathcal{T}\) of the full subcategory \(\mathcal{D}^{f}(R)\) of the derive category \(\mathcal{D}(R)\), consisting of \(R\)-complexes with finitely generated homologies, is ``self-dual'' under Grothendieck duality. In other words, if \(X \in \mathcal{T}\), then \(\mathrm{RHom}_{R}(X,R) \in \mathcal{T}\). It is worth noting that this result had been proved in [\textit{G. Stevenson}, Bull. Lond. Math. Soc. 46, No. 2, 245--257 (2014; Zbl 1321.13008)] under the additional assumption that \(R\) is a quotient of a regular ring modulo a regular sequence. Therefore, the authors relax this extra assumption. To prove this result, the authors employ two different approaches, the first one relying on the theory of cohomological support, and the second one using the graded Hopf algebra structure of the exterior algebra, thereby they present two proofs of their theorem. As a consequence of this theorem, the authors recover a result of Avramov and Buchweitz that says the eventual vanishing of Ext is equivalent to the eventual vanishing of Tor over locally complete intersections. Furthermore, they show that if \(R\) is local, then the polynomial rate in \(n\) of growth of the minimal number of generators of \(\mathrm{Ext}^{n}_{R}(X,Y)\) for any \(X,Y \in \mathcal{D}^{f}(R)\), the so-called complexity of \(X\) and \(Y\), is symmetric in \(X\) and \(Y\).
Reviewer: Hossein Faridian (Clemson)Resolutions of letterplace ideals of posets.https://www.zbmath.org/1453.130412021-02-27T13:50:00+00:00"D'Alì, Alessio"https://www.zbmath.org/authors/?q=ai:dali.alessio"Fløystad, Gunnar"https://www.zbmath.org/authors/?q=ai:floystad.gunnar"Nematbakhsh, Amin"https://www.zbmath.org/authors/?q=ai:nematbakhsh.aminSummary: We investigate resolutions of letterplace ideals of posets. We develop topological results to compute their multigraded Betti numbers, and to give structural results on these Betti numbers. If the poset is a union of no more than \(c\) chains, we show that the Betti numbers may be computed from simplicial complexes of no more than \(c\) vertices. We also give a recursive procedure to compute the Betti diagrams when the Hasse diagram of \(P\) has tree structure.A note on \(w\)-GD domains.https://www.zbmath.org/1453.130162021-02-27T13:50:00+00:00"Zhou, Dechuan"https://www.zbmath.org/authors/?q=ai:zhou.dechuanSummary: Let \(S\) and \(T\) be \(w\)-linked extension domains of a domain \(R\) with \(S\subseteq T\). In this paper, we define what satisfying the \(w_R\)-GD property for \(S \subseteq T\) means and what being \(w_R\)- or \(w\)-GD domains for \(T\) means. Then some sufficient conditions are given for the \(w_R\)-GD property and \(w_R\)-GD domains. For example, if \(T\) is \(w_R\)-integral over \(S\) and \(S\) is integrally closed, then the \(w_R\)-GD property holds. It is also given that \(S\) is a \(w_R\)-GD domain if and only if \(S\subseteq T\) satisfies the \(w_R\)-GD property for each \(w_R\)-linked valuation overring \(T\) of \(S\), if and only if \(S\subseteq (S[u])_w\) satisfies the \(w_R\)-GD property for each element \(u\) in the quotient field of \(S\), if and only if \(S_{\mathfrak{m}}\) is a GD domain for each maximal \(w_R\)-ideal \(\mathfrak{m}\) of \(S\). Then we focus on discussing the relationship among GD domains, \(w\)-GD domains, \(w_R\)-GD domains, Prüfer domains, \(Pv\) MDs and \(Pw_R\) MDs, and also provide some relevant counterexamples. As an application, we give a new characterization of \(Pw_R\) MDs. We show that \(S\) is a \(Pw_R\) MD if and only if \(S\) is a \(w_R\)-GD domain and every \(w_R\)-linked overring of \(S\) that satisfies the \(w_R\)-GD property is \(w_R\)-flat over \(S\). Furthermore, examples are provided to show these two conditions are necessary for \(Pw_R\) MDs.The number of star operations on numerical semigroups and on related integral domains.https://www.zbmath.org/1453.200812021-02-27T13:50:00+00:00"Spirito, Dario"https://www.zbmath.org/authors/?q=ai:spirito.darioThe author studies the cardinality of the set \(\mathrm{Star}(S)\) of star operations on a numerical semigroup \(S\). In particular, he studies ways to estimate \(\mathrm{Star}(S)\) and to bound the number of nonsymmetric numerical semigroups such that \(|\mathrm{Star}(S)|\leq n\). He gives a rather precise asymptotic expression for the number of semigroups of multiplicity 3 with less than \(n\) star operations (Theorem 6.4), and a bound for the semigroups of prime multiplicity (Theorem 7.4). He lists all nonsymmetric numerical semigroups with 150 or less star operations (Table 4), and proves an explicit bound for residually
rational rings (Theorem 10.5). The paper contains ten sections. In Sections 2 and 3, he presents basic material.
Sections 4 and 5 present estimates already presented in [\textit{D. Spirito}, Commun. Algebra 43, No. 7, 2943--2963 (2015; Zbl 1339.13013); J. Commut. Algebra 11, No. 3, 401--431 (2019; Zbl 1450.20020)]. Section 6 deepens the analysis of \textit{D. Spirito} [Semigroup Forum 91, No. 2, 476--494 (2015; Zbl 1350.20045)] on semigroups of multiplicity 3, and Section 7 studies the case where the multiplicity is prime (and bigger than 3). Section 8 introduces the concept of linear families and Section 9 is devoted to algorithms to calculate \(|\mathrm{Star}(S)|\) and to determine all the nonsymmetric semigroups with at most \(n\) star operations. The last section, Section 10, studies the domain case and contains analogues of the results of Section 4 for residually rational domains.
For the entire collection see [Zbl 1446.20006].
Reviewer: A. Mimouni (Dhahran)Frobenius actions on local cohomology modules and deformation.https://www.zbmath.org/1453.130212021-02-27T13:50:00+00:00"Ma, Linquan"https://www.zbmath.org/authors/?q=ai:ma.linquan"Quy, Pham Hung"https://www.zbmath.org/authors/?q=ai:pham-hung-quy.A property \(P\) on a ring is said to deform if \(R/(x)\) has property \(P\) for some non-zerodivisor \(x\) implies that \(R\) has property \(P\). Of late, much progress has been made in the study of deformation of various \(F\)-singularities. In this paper, the authors focus mainly on deformation of the three \(F\)-singularities: \(F\)-full (\(R\) is \(F\)-full if and only if \(F_*(H^i_{\mathfrak{m}}(R) )\rightarrow H^i_{\mathfrak{m}}(R)\) is surjective for all \(i\geq 0\)), \(F\)-anti-nilpotent (\(R\) is \(F\)-anti-nilpotent if for all \(i \geq 0\) and all \(F\)-stable submodules \(N \subseteq H^i_{\mathfrak{m}}(R)\), the Frobenius action is injective on \(H^i_{\mathfrak{m}}(R)/N\)) and \(F\)-injective (\(R\) is \(F\)-injective if the Frobenius action is injective on \(H^i_{\mathfrak{m}}(R)\) for all \(i \geq 0\)). Several examples and remarks clarify the relationship between these \(F\)-singularities. The authors show that both \(F\)-fullness and \(F\)-anti-nilpotence deform for local rings of positive characteristic. They further show that if \(R\) is a local ring of characteristic \(p\) with perfect residue field and if there exists a regular element \(x\) such that the cokernel of \(\phi_x :H^i_{\mathfrak{m}}(R) \rightarrow H^i_{\mathfrak{m}}(R)\) has finite length for all \(i \geq 0\) where \(\phi_x(\eta)=x\eta\), then \(F\)-injectivity deforms.
Reviewer: Janet Vassilev (Albuquerque)Some finiteness results for co-associated primes of generalized local homology modules and applications.https://www.zbmath.org/1453.130512021-02-27T13:50:00+00:00"Do, Yen Ngoc"https://www.zbmath.org/authors/?q=ai:do.yen-ngoc"Nguyen, Tri Minh"https://www.zbmath.org/authors/?q=ai:nguyen.tri-minh"Tran, Nam Tuan"https://www.zbmath.org/authors/?q=ai:tran.nam-tuanSummary: We prove some results about the finiteness of co-associated primes of generalized local homology modules inspired by a conjecture of \textit{A. Grothendieck} [Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz loceaux et globeaux (SGA 2). Augmenté d'un exposé par Michèle Raynaud. Séminaire de géométrie algébrique du Bois-Marie 1962. Amsterdam: North-Holland Publishing Company; Paris: Masson \& Cie (1968; Zbl 0197.47202)] and a question of \textit{C. Huneke} [Res. Notes Math. 2, 93--108 (1992; Zbl 0782.13015)]. We also show some equivalent properties of minimax local homology modules. By duality, we get some properties of Herzog's generalized local cohomology modules.Resolving decompositions for polynomial modules.https://www.zbmath.org/1453.130392021-02-27T13:50:00+00:00"Albert, Mario"https://www.zbmath.org/authors/?q=ai:albert.mario"Seiler, Werner M."https://www.zbmath.org/authors/?q=ai:seiler.werner-mSummary: We introduce the novel concept of a resolving decomposition of a polynomial module as a combinatorial structure that allows for the effective construction of free resolutions. It provides a unifying framework for recent results of the authors for different types of bases.
For the entire collection see [Zbl 1346.68010].On trace of symmetric bi-derivations on rings.https://www.zbmath.org/1453.160432021-02-27T13:50:00+00:00"Leerawat, Utsanee"https://www.zbmath.org/authors/?q=ai:leerawat.utsanee"Khun-in, Sitanan"https://www.zbmath.org/authors/?q=ai:khun-in.sitananSummary: The purpose of this paper is to prove some results concerning symmetric bi-derivations on prime and semiprime rings. Moreover, we characterize symmetric bi-derivations when their traces satisfies certain conditions on prime and semiprime rings. Furthermore, the commutativity of a prime ring satisfying certain identity involving symmetric bi-derivations of a prime ring is discussed.A short biography of Professor Leonid Bokut.https://www.zbmath.org/1453.130022021-02-27T13:50:00+00:00"Kolesnikov, P."https://www.zbmath.org/authors/?q=ai:kolesnikov.pavel-sThe range of all regularities for polynomial ideals with a given Hilbert function.https://www.zbmath.org/1453.130782021-02-27T13:50:00+00:00"Cioffi, Francesca"https://www.zbmath.org/authors/?q=ai:cioffi.francescaLet \(A\) be a polynomial ring over a field \(K\) and let \(I\) denote any homogeneous ideal of \(A\). This paper presents a proof of the following statement suggested by Le Tuan Hoa in a private communication,
Theorem 1: Given an Hilbert function \(u\), the set \(\{reg(I) : A/I \text{ has Hilbert function } u \}\) is an interval of integers.
Since for every homogenous ideal \(I\), there exists a strongly stable ideal \(J\) such that \(reg(I)=reg(J)\) and \(reg(J)\) is the highest degree, denoted by \(\nu(J)\), of a minimal generator of \(J\), the theorem 1 is equivalent to the following one,
Theorem 2: Given an Hilbert function \(u\), the set \(\{\nu(J) : J \text{ is strongly stable and }A/J \text{ has Hilbert function } u\}\) is an interval of integers.
The proof is based on constructive arguments presented in the paper [\textit{F. Cioffi} et al., Exp. Math. 24, No. 4, 424--437 (2015; Zbl 1333.13028)], mainly on a construction called \textit{expanded lifting}, that allows, given a saturated homogeneous ideal \(I\) with Hilbert function \(\preceq u\), to create a homogenous ideal \(J\) with Hilbert function \(u\).
Reviewer: Gema Maria Diaz Toca (Murcia)The \(B\)-model connection and mirror symmetry for Grassmannians.https://www.zbmath.org/1453.141042021-02-27T13:50:00+00:00"Marsh, R. J."https://www.zbmath.org/authors/?q=ai:marsh.robert-j"Rietsch, K."https://www.zbmath.org/authors/?q=ai:rietsch.konstanzeSummary: We consider the Grassmannian \(X = G r_{n - k}(\mathbb{C}^n)\) and describe a `mirror dual' Landau-Ginzburg model \((\check{\mathbb{X}}^\circ, W_q : \check{\mathbb{X}}^\circ \to \mathbb{C})\), where \(\check{\mathbb{X}}^\circ\) is the complement of a particular anti-canonical divisor in a Langlands dual Grassmannian \(\check{\mathbb{X}}\), and we express \(W\) succinctly in terms of Plücker coordinates. First of all, we show this Landau-Ginzburg model to be isomorphic to one proposed for homogeneous spaces in a previous work by the second author. Secondly we show it to be a partial compactification of the Landau-Ginzburg model defined in the 1990's by \textit{T. Eguchi} et al. [Int. J. Mod. Phys. A 12, No. 9, 1743--1782 (1997; Zbl 1072.32500)]. Finally we construct inside the Gauss-Manin system associated to \(W_q\) a free submodule which recovers the trivial vector bundle with small Dubrovin connection defined out of Gromov-Witten invariants of \(X\). We also prove a \(T\)-equivariant version of this isomorphism of connections. Our results imply in the case of Grassmannians an integral formula for a solution to the quantum cohomology \(D\)-module of a homogeneous space, which was conjectured by the second author. They also imply a series expansion of the top term in Givental's \(J\)-function, which was conjectured in [\textit{V. V. Batyrev} et al., Nucl. Phys., B 514, No. 3, 640--666 (1998; Zbl 0896.14025)].Presentation of the book The Gröbner cover.https://www.zbmath.org/1453.130832021-02-27T13:50:00+00:00"Montes, Antonio"https://www.zbmath.org/authors/?q=ai:montes.antonioSummary: The aim of this paper is to present the research book [the author, The Gröbner cover. Cham: Springer (2018; Zbl 1412.13001)] recently published. This book is divided into two parts, one theoretical and one focusing on applications, and offers a complete description of the Canonical Gröbner Cover, to the author's best knowledge, the most accurate algebraic method for discussing parametric polynomial systems. It also includes applications to the automatic deduction of geometric theorems, loci computation and envelopes. The theoretical part is a self-contained exposition on the theory of Parametric Gröbner Systems and Bases. It begins with Weispfenning introduction of Comprehensive Gröbner Systems (CGS), a fundamental contribution made in 1992, and provides a complete description of the Canonical Gröbner Cover (GC), which includes a canonical discussion of a set of parametric polynomial equations developed in [the author and \textit{M. Wibmer}, J. Symb. Comput. 45, No. 12, 1391--1425 (2010; Zbl 1207.13018)]. In turn, the application part selects three problems for which the Gröbner Cover offers valuable new perspectives. The automatic deduction of geometric theorems (ADGT) becomes fully automatic and straightforward using GC, representing a major improvement on all previous methods. In terms of loci and envelope computation, GC makes it possible to introduce a taxonomy of the components and automatically compute it. The book also generalizes the definition of the envelope of a family of hyper-surfaces, and provides algorithms for its computation, as well as for discussing how to determine the real envelope. All the algorithms described in the book have also been included in the Singular software library grobcov.lib implemented by the author and H. Schönemann, the book serving also as User Manual for the library.Categorified duality in Boij-Söderberg theory and invariants of free complexes.https://www.zbmath.org/1453.130422021-02-27T13:50:00+00:00"Eisenbud, David"https://www.zbmath.org/authors/?q=ai:eisenbud.david"Erman, Daniel"https://www.zbmath.org/authors/?q=ai:erman.danielSummary: We present a robust categorical foundation for the duality theory introduced by Eisenbud and Schreyer to prove the Boij-Söderberg conjectures describing numerical invariants of syzygies. The new foundation allows us to extend the reach of the theory substantially.
More explicitly, we construct a pairing between derived categories that simultaneously categorifies all the functionals used by Eisenbud and Schreyer. With this new tool, we describe the cone of Betti tables of finite, minimal free complexes having homology modules of specified dimensions over a polynomial ring, and we treat many examples beyond polynomial rings. We also construct an analogue of our pairing between derived categories on a toric variety, yielding toric/multigraded analogues of the Eisenbud-Schreyer functionals.On Rees algebras of linearly presented ideals in three variables.https://www.zbmath.org/1453.130182021-02-27T13:50:00+00:00"Lan, Nguyen P. H."https://www.zbmath.org/authors/?q=ai:lan.nguyen-p-hSummary: Let \(I\) be a height two perfect ideal with a linear presentation matrix in a polynomial ring \(R = k [x, y, z]\). Assume furthermore that after modulo an ideal generated by two variables, the presentation matrix has rank one. We describe the defining ideal of the Rees algebra \(\mathcal{R}(I)\) explicitly and we show that \(\mathcal{R}(I)\) is Cohen-Macaulay.Exploration of dual curves using a dynamic geometry and computer algebra system.https://www.zbmath.org/1453.682232021-02-27T13:50:00+00:00"Hašek, Roman"https://www.zbmath.org/authors/?q=ai:hasek.romanSummary: This submission presents a particular example of the use of the free dynamic mathematics software GeoGebra to determine the dual curve to the given curve and to inspect its properties. The example is aimed at students of mathematics teaching. The combination of dynamic geometry tools with computer algebra functions, namely the functions for the computation of Groebner bases for polynomial ideals and a function for eliminating variables from the system of polynomial equations, based also on the method of Groebner bases, allows them to apply naturally the knowledge they acquire during their study of mathematics.When are \(n\)-syzygy modules \(n\)-torsionfree?https://www.zbmath.org/1453.130442021-02-27T13:50:00+00:00"Matsui, Hiroki"https://www.zbmath.org/authors/?q=ai:matsui.hiroki"Takahashi, Ryo"https://www.zbmath.org/authors/?q=ai:takahashi.ryo"Tsuchiya, Yoshinao"https://www.zbmath.org/authors/?q=ai:tsuchiya.yoshinaoSummary: Let \(R\) be a commutative Noetherian ring. We consider the question of when \(n\)-syzygy modules over \(R\) are \(n\)-torsionfree in the sense of \textit{M. Auslander} and \textit{M. Bridger} [Stable module theory. Providence, RI: American Mathematical Society (AMS) (1969; Zbl 0204.36402)]. Our tools include Serre's condition and certain conditions on the local Gorenstein property of \(R\). Our main result implies the converse of a celebrated theorem of \textit{E. G. Evans} and \textit{P. Griffith} [Syzygies. Cambridge: Cambridge University Press (1985; Zbl 0569.13005)].Rings with canonical reductions.https://www.zbmath.org/1453.130692021-02-27T13:50:00+00:00"Rahimi, Mehran"https://www.zbmath.org/authors/?q=ai:rahimi.mehranLet \((R,\mathfrak{m})\) be a Noethrian local ring. For \(R\)-ideals \(J \subseteq I\), \(J\) is said to be a reduction of \(I\) if \(JI^n = I^{n+1}\) for all \(n\) large. Over a Cohen-Macaulay local ring \((R,\mathfrak{m})\) of dimension \(d\), a maximal Cohen-Macaulay module \(\omega\) is called a canonical module of \(R\) if \(\dim_k \text{Ext}^i_R(k, \omega) = \delta_{id}.\) An ideal of \(R\), isomorphic to a canonical module, is called a canonical ideal of \(R.\) In the paper under review, the authors define canonical reductions and study class of Cohen-Macaulay local rings with canonical reductions.
Let \((R,\mathfrak{m})\) be a one-dimensional Cohen-Macaulay local ring. An ideal \(I\) is called a canonical reduction of \(R\) if \(I\) is a canonical ideal of \(R\) and is a reduction of \(\mathfrak{m}.\) It is known that a local ring admits a canonical module if and only if it is a homomorphic image of a Gorenstein ring. Such a ring \(R\) admits a canonical ideal if and only if \(R_\mathfrak{p}\) is Gorenstein for all associated prime ideals \(\mathfrak{p}\) of \(R.\) Therefore, one-dimensional Gorenstein local rings, with infinite residue fields, admit canonical reductions. In case of non-Gorenstein Cohen-Macaulay local rings of dimension one, the authors prove that if a canonical reduction \(J\) exists, then it is a maximal canonical ideal which is not contained in \(\mathfrak{m}^2\) and \(\ell(R/J) = \min\{ \ell(R/K) \mid K \text{ is a canonical ideal of } R \}.\) The authors also give a class of rings that do not have a canonical reduction. They show that non-almost Gorenstein rings with minimal multiplicity do not have a canonical reduction. A characterization for a numerical semigroup ring to admit a canonical reduction is provided and the behavior of certain Ulrich ideals in a ring with canonical reduction is also studied.
Let \(R\) be a one-dimensional Cohen-Macaulay local ring with canonical module \(K_R\) and let \(M\) be a maximal Cohen-Macaulay \(R\)-module. The following statements are proved equivalent:
(1) The idealization \(R \ltimes M\) admits a canonical reduction \(I \ltimes L\), for some submodule \(L\) of \(M.\)
(2) \(M \simeq \text{Hom}_R(I,K_R)\) and \(I\) is a reduction of \(\mathfrak{m}.\)
Proceeding further with this study, the authors show a nice behavior of rings with canonical reductions via idealization. They prove that the above ring \(R\) has a canonical reduction iff \(R \ltimes R\) has a canonical reduction iff \(R \ltimes \mathfrak{m}\) has a canonical reduction.
In rings with dimension \(d >1\), the canonical reduction is defined as follows:
Assume that \((R,\mathfrak{m})\) is a \(d\)-dimensional Cohen-Macaulay local ring admitting a canonical ideal. The ring \(R\) is said to have a canonical reduction \(K\) if \(K\) is a canonical ideal of \(R\) and there exists an equimultiple ideal \(I\), such that \(\text{ht}(I) = d-1\) and \(K + I\) is a reduction of \(\mathfrak{m}.\)
It is proved that every nearly Gorenstein ring admits a canonical reduction. A criterion for a generalized Gorenstein ring to have a canonical reduction is also given. In particular, it shows that every \(d\)-dimensional almost Gorenstein ring admits a canonical reduction. Finally, the authors study the canonical reduction via linkage and flat local homomorphisms.
Reviewer: Kriti Goel (Mumbai)Characterizations of freeness for equidimensional subspaces.https://www.zbmath.org/1453.140132021-02-27T13:50:00+00:00"Pol, Delphine"https://www.zbmath.org/authors/?q=ai:pol.delphineSummary: The purpose of this paper is to investigate properties of the minimal free resolution of the modules of multi-logarithmic forms along a reduced equidimensional subspace. We first consider a notion of freeness for reduced complete intersections, and more generally for reduced equidimensional subspaces embedded in a smooth manifold, which generalizes the notion of Saito free divisors. The first main result is a characterization of freeness in terms of the projective dimension of the module of multi-logarithmic \(k\)-forms, where \(k\) is the codimension.
We also prove that there is a perfect pairing between the module of multi-logarithmic differential \(k\)-forms and the module of multi-logarithmic \(k\)-vector fields which generalizes the duality between the corresponding modules in the hypersurface case. We deduce from this perfect pairing a duality between the Jacobian ideal and the module of multi-residues of multi-logarithmic \(k\)-forms.
In the last part of this paper, we investigate logarithmic modules along some examples of free singularities. The main result in this section is an explicit computation of the minimal free resolution of the module of multi-logarithmic forms and multi-residues for quasi-homogeneous complete intersection curves.An analytical proof of the general composition theorem for formal power series and matrix representation.https://www.zbmath.org/1453.130612021-02-27T13:50:00+00:00"Gan, Xiao-Xiong"https://www.zbmath.org/authors/?q=ai:gan.xiao-xiongThe general composition theorem gives a necessary and sufficient condition for the composition of formal power series over a field with metric. It has been studied by many authors. See for example [\textit{X. Gan} and \textit{N. Knox}, Int. J. Math. Math. Sci. 30, No. 12, 761--770
(2002; Zbl 0998.13010); \textit{M. Borkowski} and \textit{P. Mackowiak}, Commentat. Math. Univ. Carol. 53, No. 4, 549--555 (2012; Zbl 1274.13040)] or [\textit{X. Gan}, Int. J. Evol. Equ. 10, No. 1, 63--74 (2015; Zbl 1360.13051)]. In this paper under review the author gives an analytical approach for the composition of formal power series. This new approach uses basically, for a formal power series \(f,\) the multinomial co-factors of \(f^{n}\) in order to write the different coefficients of \(f^{n}\). Moreover, the author provides a matrix representation for the general composition of formal power series.
Reviewer: Sana Hizem (Monastir)Correction to: Depth functions of symbolic powers of homogeneous ideals.https://www.zbmath.org/1453.130352021-02-27T13:50:00+00:00"Nguyen, Hop Dang"https://www.zbmath.org/authors/?q=ai:nguyen.dang-hop"Trung, Ngo Viet"https://www.zbmath.org/authors/?q=ai:ngo-viet-trung.Summary: The original proof of Theorem 3.3 in [\textit{H. D. Nguyen} and \textit{N. V. Trung}, Invent. Math. 218, No. 3, 779--827 (2019; Zbl 1434.13014)] incorrect. The correction concerns only this proof and does not affect any result of the paper.On graphs with 2 trivial distance ideals.https://www.zbmath.org/1453.050432021-02-27T13:50:00+00:00"Alfaro, Carlos A."https://www.zbmath.org/authors/?q=ai:alfaro.carlos-aDistance ideals generalize the Smith normal form of the distance matrix of a graph. The family of graphs with 2 trivial distance ideals contains the family of graphs whose distance matrix has at most 2 invariant factors equal to 1. Here, the author gives an infinite family of forbidden induced subgraphs for the graphs with 2 trivial distance ideals. They are also related with other well known graph classes.
Let \(G=(V,E)\) be a connected graph and \(X_G=\{x_u : u \in V(G)\}\) a set of indeterminates. The distance \(d_G(u,v)\) between the vertices \(u\) and \(v\) is the number of edges of a shortest path between them. Let \(\operatorname{diag}(X_G)\) denote the diagonal matrix with the indeterminates in the diagonal. The distance matrix \(D(G)\) of \(G\) is the matrix with rows and columns indexed by the vertices of \(G\) where the \(uv\)-entry is equal to \(d_G(u,v)\). Thus the generalized distance matrix \(D(G,X_G)\) of \(G\) is the matrix with rows and columns indexed by the vertices of \(G\) defined as \(\operatorname{diag}(X_G)+D(G)\). Note that one can recover the distance matrix from the generalized distance matrix by evaluating \(X_G\) at the zero vector, that is, \(D(G) = D(G,0)\).
Let \(R[X_G]\) be the polynomial ring over a commutative ring \(R\) in the variables \(X_G\). For all \(i \in \lfloor n\rfloor := \{1,2,\dots,n\}\), the \(i\)-th distance ideal \(I^{R}_{i}(G,X_G)\) of \(G\) is the ideal over \(R[X_G]\) given by \(\langle\mathrm{minors}_i(D(G,X_G))\rangle\), where \(n\) is the number of vertices of \(G\) and \(\mathrm{minors}_i(D(G,X_G))\) is the set of the determinants of the \(i \times i\) submatrices of \(D(G,X_G)\).
Distance ideals were defined by \textit{C. A. Alfaro} and \textit{L. Taylor} [Linear Algebra Appl. 584, 127--144 (2020; Zbl 1426.05060)] as a generalization of the Smith normal form of distance matrix and the distance spectra of graphs. In this paper, the discussion is held in the case when \(R\) is the ring of integers of numbers.
Smith normal forms have been useful in understanding algebraic properties of combinatorial objects. For instance, computing the Smith normal form of the adjacency or Laplacian matrix is a standard technique used to determine the Smith group and the critical group of a graph.
Smith normal forms can be computed using rows and column operations. In fact, \textit{R. Kannan} and \textit{A. Bachem} [SIAM J. Comput. 8, 499--507 (1979; Zbl 0446.65015)] found polynomial algorithms for computing the Smith normal form of an integer matrix. An alternative way of obtaining the Smith normal form is as follows. Let \(\Delta_i(G)\) denote the greatest common divisor of the \(i\)-minors of the distance matrix \(D(G)\), then its \(i\)-th invariant factor \(f_i\) is equal to \(\Delta_i(G)/\Delta_{i-1}(G)\), where \(\Delta_0(G) = 1\). Thus the Smith normal form of \(D(G)\) is equal to \(\operatorname{diag}(f_1,f_2,\dots,f_r,0,\dots,0)\). It is known that the Smith normal form may not exist in ring \(\mathbb{Z}[X]\), for example consider the following matrix with entries in the ring \(\mathbb{Z}[x]\)
\[
\begin{bmatrix}
2 & 0 \\
0 & x \\
\end{bmatrix}
\quad.
\]
Little is known about the Smith normal forms of distance matrices. In the paper by \textit{Y. Hou} and \textit{C. Woo} [Linear Multilinear Algebra 56, No. 6, 611--626 (2008; Zbl 1149.05033)], the Smith normal forms of the distance matrices were determined for trees, wheels, cycles, and complements of cycles and are partially determined for complete multipartite graphs. In the paper by \textit{R. B. Bapat} and \textit{M. Karimi} [ibid. 65, No. 6, 1117--1130 (2017; Zbl 1360.05094)], the Smith normal form of the distance matrices of unicyclic graphs and of the wheel graph with trees attached to each vertex were obtained.
An ideal is said to be unit or trivial if it is equal to \(\left\langle 1\right\rangle\). Let \(\Phi(G)\) denote the maximum integer \(i\) for which \(I^{\mathbb{Z}}_{i}(G,X_G)\) is trivial. Let \(\Lambda_k\) denote the family of graphs with at most \(k\) trivial distance ideals over \(\mathbb{Z}\). Note that every graph with at least one non-loop edge has at least one trivial distance ideals. On the other hand, let \(\phi(G)\) denote the number of invariant factors of the distance matrix of \(G\) equal to 1. In this case, every graph with at least one non-loop edge has at least two invariant factors equal to one.
This paper intends to explore the properties of the family \(\Lambda_2\) of graphs with at most two trivial distance ideals over \(\mathbb{Z}\). In particular, the author has found an infinite number of graphs that are forbidden for \(\Lambda_2\). It is defined that \(F\) is the set of 17 graphs with certain properties. In Section 2, the author has proved that graphs in \(\Lambda_2\) are \(\left\{F,\text{odd-holes}\right\}\)-free graphs, where odd-holes are cycles of odd length greater or equal than 7.
One of the main application in finding a characterization of \(\Lambda_2\) is that it is an approach to obtain a characterization of the graphs with \(\phi_Z(G)=2\) since they are contained in \(\Lambda_2\). It has been proved that the distance matrix of trees has exactly 2 invariant factors equal to 1. Therefore,
\[ \text{trees} \subseteq \Lambda_2\subseteq \{F, \text{odd-holes}\}-\text{free graphs}. \]
Among the forbidden graphs for \(\Lambda_2\) there are several graphs studied in other contexts, like bull and odd-holes. Another related family is the family of 3-leaf powers that was characterized as \(\left\{\text{bull, dart, gem}\right\}\)-free chordal graphs.
Distance-hereditary graphs are another related family defined by \textit{E. Howorka} [Q. J. Math., Oxf. II. Ser. 28, 417--420 (1977; Zbl 0376.05040)]. A graph \(G\) is distance-hereditary if for each connected induced subgraph \(H\) of \(G\) and every pair \(u\) and \(v\) of vertices in \(H\), \(d_H(u,v) = d_G(u,v)\). Distance-hereditary graphs are \(\left\{\text{house, gem, domino, holes}\right\}\)-free graphs, where holes are cycles of length greater than or equal 5 which intersects with \(\Lambda_2\). Also, if \(H\) is a connected induced subgraph of a distance-hereditary graph \(G\), then \(I^{R}_{i}(H, X_H) \subseteq I^{R}_{i}(G,X_G)\) for all \(i \leq \left|V(H)\right|\).
Previously, an analogous notion to the distance ideals but for the adjacency and Laplacian matrices was explored. They are called critical ideal. It was found new connections in contexts different from the Smith group or sandpile group like the zero-forcing number and the minimum rank of a graph. In this setting, the set of forbidden graphs for the family with at most \(k\) trivial critical ideals is conjectured to be finite. It is interesting that for distance ideals this is not true.
A graph \(G\) is forbidden for \(\Lambda_k\) if the \((k+1)\)-th distance ideal of \(G\) is trivial. In addition, a forbidden graph \(H\) for \(\Lambda_k\) is minimal if \(H\) does not contain a connected forbidden graph for \(\Lambda_k\) as induced subgraph, and any graph \(G\) containing \(H\) as induced subgraph, have that \(G\) is forbidden for \(\Lambda_k\). The set of minimal forbidden graphs for \(\Lambda_k\) is denoted by \(\mathrm{Forb}_k\).
The author prove several results. The paper is well-written. The standard of the paper is good. Researchers will benefit a lot by reading this paper.
Reviewer: A. Lourdusamy (Palayamkottai)Fermat-type arrangements.https://www.zbmath.org/1453.140222021-02-27T13:50:00+00:00"Szpond, Justyna"https://www.zbmath.org/authors/?q=ai:szpond.justynaThe present paper of the author is a survey (which contains also some original results) devoted to the so-called Fermat-type arrangements of hyperplanes. These arrangements have been studied for a long time and appeared recently in connection with highly interesting algebraic problems, namely the containment problem for homogeneous ideals and the existence of unexpected hypersurfaces. In the first part of the paper the author recalls some basic properties of reflection arrangements and free arrangements of hyperplanes. In the second part of the paper the author focuses on the existence of unexpected varieties constructed with use of the Fermat-type arrangements -- the construction uses point configurations that are dual to hyperplanes in Fermat-type arrangements. Even if the author says that she failed in writing a comprehensive survey on Fermat-type arrangements and their appearances in commutative algebra and algebraic geometry it seems that the paper contains the most recent results devoted to unexpected varieties related to Fermat-type arrangements, and in this setting it can be viewed as a up-to-date source.
For the entire collection see [Zbl 07248454].
Reviewer: Piotr Pokora (Kraków)Homological and combinatorial properties of powers of cover ideals of graphs.https://www.zbmath.org/1453.130362021-02-27T13:50:00+00:00"Seyed Fakhari, S. A."https://www.zbmath.org/authors/?q=ai:seyed-fakhari.seyed-aminSummary: Over the last 25 years the study of algebraic, homological and combinatorial properties of powers of ideals has been one of the major topics in commutative algebra. In this article, we survey the recent results concerning the associated primes, regularity, depth and Stanley depth of (symbolic) powers of cover ideals of graphs.
For the entire collection see [Zbl 07248454].Licci level Stanley-Reisner ideals with height three and with type two.https://www.zbmath.org/1453.130622021-02-27T13:50:00+00:00"Rinaldo, Giancarlo"https://www.zbmath.org/authors/?q=ai:rinaldo.giancarlo"Terai, Naoki"https://www.zbmath.org/authors/?q=ai:terai.naoki"Yoshida, Ken-Ichi"https://www.zbmath.org/authors/?q=ai:yoshida.ken-ichiSummary: Via computer-aided classification we show that the following three conditions are equivalent for level* squarefree monomial ideals \(I\) with codimension 3, with Cohen-Macaulay type 2 and with \(\dim S/I \le 4\): \((1) IS_{{\mathfrak m}}\) is licci, (2) the twisted conormal module of I is Cohen-Macaulay, \((3) S/I^{(2)}\) is Cohen-Macaulay, where \(S\) is a polynomial ring over a field of characteristic 0 and \({\mathfrak m}\) is its graded maximal ideal.
For the entire collection see [Zbl 07248454].The Bass-Quillen conjecture and Swan's question.https://www.zbmath.org/1453.130262021-02-27T13:50:00+00:00"Popescu, Dorin"https://www.zbmath.org/authors/?q=ai:popescu.dorinSummary: We present a question which implies a complete positive answer for the Bass-Quillen Conjecture.
For the entire collection see [Zbl 07248454].Asymptotic behavior of symmetric ideals: a brief survey.https://www.zbmath.org/1453.130242021-02-27T13:50:00+00:00"Juhnke-Kubitzke, Martina"https://www.zbmath.org/authors/?q=ai:juhnke-kubitzke.martina"Le, Dinh Van"https://www.zbmath.org/authors/?q=ai:le.dinh-van"Römer, Tim"https://www.zbmath.org/authors/?q=ai:romer.timSummary: Recently, chains of increasing symmetric ideals have attracted considerable attention. In this note, we summarize some results and open problems concerning the asymptotic behavior of several algebraic and homological invariants along such chains, including codimension, projective dimension, Castelnuovo-Mumford regularity, and Betti tables.
For the entire collection see [Zbl 07248454].Depth of an initial ideal.https://www.zbmath.org/1453.130812021-02-27T13:50:00+00:00"Hibi, Takayuki"https://www.zbmath.org/authors/?q=ai:hibi.takayuki"Tsuchiya, Akiyoshi"https://www.zbmath.org/authors/?q=ai:tsuchiya.akiyoshiSummary: Given an arbitrary integer \(d>0\), we construct a homogeneous ideal I of the polynomial ring \(S = K[x_1, \ldots , x_{3d}]\) in 3d variables over a field K for which S/I is a Cohen-Macaulay ring of dimension \(d\) with the property that, for each of the integers \(0 \le r \le d\), there exists a monomial order \(<_r\) on \(S\) with \(\text{depth} (S/\text{in}_{<_r}(I)) = r\), where \(\text{in}_{<_r}(I)\) is the initial ideal of \(I\) with respect to \(<_r\).
For the entire collection see [Zbl 07248454].Gröbner-nice pairs of ideals.https://www.zbmath.org/1453.130772021-02-27T13:50:00+00:00"Cimpoeaş, Mircea"https://www.zbmath.org/authors/?q=ai:cimpoeas.mircea"Stamate, Dumitru I."https://www.zbmath.org/authors/?q=ai:stamate.dumitru-ioanSummary: We introduce the concept of a Gröbner nice pair of ideals in a polynomial ring and we present some applications.
For the entire collection see [Zbl 07248454].Nearly normally torsionfree ideals.https://www.zbmath.org/1453.130072021-02-27T13:50:00+00:00"Andrei-Ciobanu, Claudia"https://www.zbmath.org/authors/?q=ai:andrei-ciobanu.claudiaSummary: We describe all connected graphs whose edge ideals are nearly normally torsionfree. We also prove that the facet ideal of a special odd cycle is nearly normally torsionfree. Finally, we give a necessary condition for a t-spread principal Borel ideal generated in degree 3 to be nearly normally torsionfree.
For the entire collection see [Zbl 07248454].Syzygies of Cohen-Macaulay modules and Grothendieck groups.https://www.zbmath.org/1453.130332021-02-27T13:50:00+00:00"Kobayashi, Toshinori"https://www.zbmath.org/authors/?q=ai:kobayashi.toshinoriSummary: We study the converse of a theorem of Butler and Auslander-Reiten. We show that a Cohen-Macaulay local ring with an isolated singularity has only finitely many isomorphism classes of indecomposable summands of syzygies of Cohen-Macaulay modules if the Auslander-Reiten sequences generate the relation of the Grothendieck group of finitely generated modules. This extends a recent result of Hiramatsu, which gives an affirmative answer in the Gorenstein case to a conjecture of Auslander.Quotient structure and chain conditions on quasi modules.https://www.zbmath.org/1453.130372021-02-27T13:50:00+00:00"Jana, Sandip"https://www.zbmath.org/authors/?q=ai:jana.sandip"Mazumder, Supriyo"https://www.zbmath.org/authors/?q=ai:mazumder.supriyoSummary: Quasi module is an axiomatisation of the hyperspace structure based on a module. We initiated this structure in our paper [``An associated structure of a module'', Rev. Acad. Canar. Cienc. 25, No. 2, 9--22 (2013)]. It is a generalisation of the module structure in the sense that every module can be embedded into a quasi module and every quasi module contains a module. The structure of quasi module is a conglomeration of a commutative semigroup with an external ring multiplication and a compatible partial order. In the entire structure partial order has an intrinsic effect and plays a key role in any development of the theory of quasi modules. In the present paper we have discussed the quotient structure of a quasi module by introducing a congruence suitably. Also we introduce the concept of chain conditions on quasi modules and prove some theorems related to chain conditions.The weak Lefschetz property of Gorenstein algebras of codimension three associated to the Apéry sets.https://www.zbmath.org/1453.130592021-02-27T13:50:00+00:00"Miró-Roig, Rosa M."https://www.zbmath.org/authors/?q=ai:miro-roig.rosa-maria"Tran, Quang Hoa"https://www.zbmath.org/authors/?q=ai:tran.quang-hoa.1Let \(k\) be a field of characteristic zero. It has been conjectured that every graded Artinian Gorenstein algebra \(A\) of codimension three has the weak Lefschetz property (WLP), i.e. the property that multiplication by a general linear form defines a homomorphism of maximal rank from any component of \(A\) to the next. This has been proved for arbitrary complete intersections of codimension three, and it is known to be false for Artinian Gorenstein algebras of codimension four or more (although it remains open for complete intersections of codimension four or more). Many authors have explored different aspects of the WLP problem, for Gorenstein algebras, monomial ideals, etc. In this paper the authors consider the WLP problem for associated graded algebras \(A\) of the Apéry set of \(M\)-pure symmetric numerical semigroups generated by four natural numbers. These were shown by \textit{L. Bryant} [Commun. Algebra 38, No. 6, 2092--2128 (2010; Zbl 1203.13004)] to be graded Artinian Gorenstein algebras of codimension three, so they are a natural class of algebras to study. \textit{L. Guerrieri} [Ark. Mat. 57, No. 1, 85--106 (2019; Zbl 1419.13011)] gives a strong structure for such algebras (in particular, they are always presented by five forms of very particular kinds) which the current authors use to advantage. They do not prove that all such algebras have the WLP, but they do prove that such \(A\) have the WLP in certain broad situations. This extends earlier work by the same authors [J. Algebra 551, 209--231 (2020; Zbl 1434.14005); J. Pure Appl. Algebra 224, No. 7, Article ID 106305, 29 p. (2020; Zbl 1437.13028)].
Reviewer: Juan C. Migliore (Notre Dame)Corrigendum to: ``The Zariski topology on sets of semistar operations without finite-type assumptions''.https://www.zbmath.org/1453.130132021-02-27T13:50:00+00:00"Spirito, Dario"https://www.zbmath.org/authors/?q=ai:spirito.darioFrom the text: We correct three problems in the author's paper [ibid. 513, 27--49 (2018; Zbl 1400.13011)], pertaining to Proposition 4.2, Lemma 5.4 and Proposition 6.1. More precisely, the proof of Proposition 4.2 is incomplete (and the proposition itself is likely wrong); the proofs of Lemma 5.4 and Proposition 6.1 both contain an error, and we fix them.A stable version of Harbourne's conjecture and the containment problem for space monomial curves.https://www.zbmath.org/1453.130092021-02-27T13:50:00+00:00"Grifo, Eloísa"https://www.zbmath.org/authors/?q=ai:grifo.eloisaIn the paper under review, the author studies the so-called containment problem. Given a radical ideal \(I\) in a domain \(R\), the \(n\)-th symbolic power of \(I\) is the ideal given by
\[I^{(n)} = \bigcap_{P \in \mathrm{Min}(I)} (I^{n}R_{P}\cap R).\]
This is the intersection of the minimal components of the ordinary power \(I^{n}\), where minimal stands for non-embedded rather than height minimal. If \(R\) is a polynomial ring, \(I^{(n)}\) is the set of functions that vanish up to order \(n\) on the variety defined by \(I\). The containment problem deals with the statements of the form \(I^{(a)} \subseteq I^{b}\): given an ideal \(I\) and a value \(b\), one would like to determine the smallest \(a\) for which \(I^{(a)} \subseteq I^{b}\). There are two main questions that build up the core of the paper under review.
Question A (Huneke). If \(P\) is a prime of height \(2\) in a regular local ring, is \(P^{(3)} \subseteq P^2\)?
The question above formulated around 2000 was later generalized by Harbourne.
Question B (Harbourne). Let \(I\) be a radical ideal of big height \(h\) in a regular ring \(R\). Then for all \(n\geq 1\), one has
\[I^{(hn - h+1)} \subseteq I^{n}.\]
The first main result of the paper under review answers on Question A positively for a certain class of ideals.
Theorem A. Let \(k\) be a field of characteristic different than \(3\), and consider \(R = k[[x,y,z]]\) or \(R = k[x,y,z]\). Let \(P\) be the prime ideal in \(R\) defining the space monomial curve \(x=t^a\), \(y=t^b\), and \(z=t^c\). Then
\[P^{(3)} \subseteq P^2.\]
The second main result proves the so-called stable version of Harbourne's conjecture.
Theorem B. Let \(R\) be a regular ring containing a field, and let \(I\) be a radical ideal In \(R\) with big height \(h\). If
\[I^{(hm-h)} \subset I^{m}\]
for some \(m\geq 2\), then
\[I^{(hk - h)} \subseteq I^{k}\]
for all \(k \geq hm\).
Reviewer: Piotr Pokora (Kraków)