Recent zbMATH articles in MSC 12https://www.zbmath.org/atom/cc/122021-04-16T16:22:00+00:00WerkzeugMultivariate difference-differential dimension polynomials.https://www.zbmath.org/1456.120032021-04-16T16:22:00+00:00"Levin, Alexander"https://www.zbmath.org/authors/?q=ai:levin.alexander-bSee the combined review about four papers by Alexander Levin (one coauthored with Alexander Egrafov) in Zbl 1456.12002.
\begin{itemize}
\item I: \textit{A. Levin}, Math. Comput. Sci. 13, No. 1--2, 157--168 (2019);
\item II: \textit{A. Levin}, J. Symb. Comput. 102, 173--188 (2021; Zbl 1456.12004);
\item III: \textit{A. Levin}, Math. Comput. Sci. 14, No. 2, 361--374 (2020);
\item IV: \textit{A. Evgrafov} and \textit{A. Levin}, Math. Comput. Sci. 14, No. 2, 347--360 (2020; Zbl 07225423).
\end{itemize}
Reviewer: Jacques Sauloy (Toulouse)Dimension polynomials and the Einstein's strength of some systems of quasi-linear algebraic difference equations.https://www.zbmath.org/1456.120012021-04-16T16:22:00+00:00"Evgrafov, Alexander"https://www.zbmath.org/authors/?q=ai:evgrafov.alexander"Levin, Alexander"https://www.zbmath.org/authors/?q=ai:levin.alexander-bSee the combined review about four papers by Alexander Levin (one coauthored with Alexander Egrafov) in Zbl 1456.12002.
\begin{itemize}
\item I: \textit{A. Levin}, Math. Comput. Sci. 13, No. 1--2, 157--168 (2019);
\item II: \textit{A. Levin}, J. Symb. Comput. 102, 173--188 (2021; Zbl 1456.12004);
\item III: \textit{A. Levin}, Math. Comput. Sci. 14, No. 2, 361--374 (2020; Zbl 1456.12003)
\item IV: \textit{A. Evgrafov} and \textit{A. Levin}, Math. Comput. Sci. 14, No. 2, 347--360 (2020).
\end{itemize}
Reviewer: Jacques Sauloy (Toulouse)Irreducible factors of a class of permutation polynomials.https://www.zbmath.org/1456.112292021-04-16T16:22:00+00:00"Kalaycı, Tekgül"https://www.zbmath.org/authors/?q=ai:kalayci.tekgul"Stichtenoth, Henning"https://www.zbmath.org/authors/?q=ai:stichtenoth.henning"Topuzoğlu, Alev"https://www.zbmath.org/authors/?q=ai:topuzoglu.alevThe authors present results on the degrees of the irreducible factors of a permutation polynomial of the finite field \({\mathbb F}_q\) of \(q\) elements.
More precisely, they study permutation polynomials \(f_n(x)\) recursively defined by
\[f_0(x)=ax+a_0,\quad f_i(x)=f_{i-1}^{d_i}(x)+a_i,~i=1,\ldots,n,\]
where \(a\in {\mathbb F}_q^*\), \(a_0,\ldots,a_n\in {\mathbb F}_q\) and \(d_1,\ldots,d_n\ge 2\) with \(\gcd(d_i,q-1)=1\), \(i=1,\ldots,n\).
Note that each permutation polynomial is of this form with \(d_i=q-2\), \(i=1,\ldots,n\), and some \(n\) by a well-known result of \textit{L. Carlitz} [Proc. Am. Math. Soc. 11, 456--459 (1960; Zbl 0095.03003)].
The first main result (Theorem 2.2) is the following. Let \(d=\mathrm{lcm}(d_1,\ldots,d_n)\) and assume \(\gcd(d,q)=1\). Then the degree of each irreducible factor \(Q(x)\) of \(F_n(x)\) is a divisor of \(d_1 d_2 \cdots d_{n-1}\mathrm{ord}_d(q)\).
Moreover, (Theorem 2.3) the degree of \(Q(x)\) is either \(1\) or divisible by ord\(_\ell(q)\) for some prime divisor \(\ell\) of \(d\). Several additional results are proved.
The results of this paper enable one to produce families of permutation polynomials of large degrees, where possible degrees of their irreducible factors are known. See also the follow-up paper of the same authors [ `` Permutation polynomials and factorization'', Cryptogr. Commun. 12, No. 5, 913--934 (2020; \url{doi:10.1007/s12095-020-00446-y})], in particular, in view of applications.
Reviewer: Arne Winterhof (Linz)Eliminating tame ramification generalizations of Abhyankar's lemma.https://www.zbmath.org/1456.120052021-04-16T16:22:00+00:00"Dutta, Arpan"https://www.zbmath.org/authors/?q=ai:dutta.arpan"Kuhlmann, Franz-Viktor"https://www.zbmath.org/authors/?q=ai:kuhlmann.franz-viktorBefore stating the theorems which are proved in this paper, we have to recall some properties of valued fields. For every valued field \((K,v)\), \(v\) extends to the algebraic closure \(K^{ac}\) of \(K\). Now \((K,v)\) is henselian if \(v\) extends in a unique way to every algebraic extension. Every valued field \((K,v)\) embeds in a minimal henselian valued field \((K^h,v)\), which is uniquely determined up to isomorphism and is called its Hensel closure. The extension \((K^h|K,v)\) is immediate, i.e. \(vK^h=vK\) and \(K^hv=Kv\), where \(vK\) denotes the valuation group of \((K,v)\) and \(Kv\) its residue field.
An algebraic extension \((L|K,v)\) of henselian valued fields is tame if, for every finite subextension \((E|K,v)\), the characteristic of \(Kv\) doesn't divide \((vE:vK)\), \(Ev|Kv\) is separable and \([E:K]=(vE:vK)[Ev:Kv]\). Next, we have the following sequence of valued fields:
\(K\subseteq K^h\subseteq K^i\subseteq K^r\subseteq K^s\subseteq K^{ac}\).
Here \(K^s\) is the separable closure of \(K\), \(K^r\) is called its ramification field and \(K^i\) its inertia field. These fields satisfy the following properties:
\(vK^i=vK\), \(K^iv=K^rv\), the extensions \(K^iv|Kv\), \(K^r|K\) and \(K^i|K\) are Galois,
\(\mathrm{Gal}(K^iv|Kv)\simeq \mathrm{Gal}(K^i|K^h)\), and if the characteristic of \(Kv\) is \(p>0\), then \(\mathrm{Gal}(K^s|K^r)\) is a pro-\(p\)-group, \(\mathrm{Gal}(K^r|K^i)\) doesn't contain any element of order \(p\) and \(K^sv|K^iv\) is purely inseparable.
In the following the extension \(L|K\) is algebraic, and \(F|K\) is an arbitrary extension.
First, assume that \(L\subseteq K^r\). The authors prove that \(LF\subseteq F^r\), \(v(LF)=vL+vF\). Further, \(LF\subseteq F^i\Leftrightarrow vL\subseteq vF\). Next, assume that \(L\subseteq K^i\). Then \(LF\subseteq F^i\), \((LF)v=Lv\cdot Fv\). Furthermore, \(LF\subseteq F^h\Leftrightarrow Lv\subseteq Fv\).
The authors deduce that if the rational rank of \(vK\) is \(1\), \((L\cdot K^h|K^h,v)\) is tame, \((vL:vK)\) and \((vF:vK)\) are finite, then \((v(LF):vL)\) is the lcm of \((vL:vK)\) and \((vF:vK)\). In particular, \(v(LF)=vF\) if, and only if, \((vL:vK)\) divides \((vF:vK)\) (\(\Leftarrow\) is an Abhyankar's lemma).
At the end of the paper they construct an example which shows that this result fails for rational rank \(2\).
In the next theorem, they assume that \(L|K\) is a normal extension and that the characteristic of \(Kv\) is \(p>0\). The greatest subgroup of \((vL)\) which contains \(vK\) and such that \(p\) doesn't divide the order of any non zero element modulo \(vK\) is denoted by \((vL)_{p'}\). First they prove that the quotient group \(v(LF)/((vL)_{p'}+vF)\) is a \(p\)-group, and they deduce that \(v(LF)/vF\) is a \(p\)-group if, and only if, \((vL)_{p'}\subseteq vF\). Next, they assume that \((vL)_{p'}=vK\) and they show that the maximal separable subextension of \((LF)v|(Lv)^sFv\) is a \(p\)-extension.
They give an example where the previous extension is nontrivial, although \(Lv=(Lv)^s=Kv\).
They also gives examples of several situations. We can quote a case where \(K=L^d=L^i\subsetneq L^r=L\), but \(F=(FL)^h\subsetneq (FL)^i=(FL)^r=FL\); so \(F=FL^i\subsetneq (FL)^i\) and \(F\subsetneq L^rF=(FL)^i\).
In another example we have \(K=L^h\subsetneq L^i=L^r=L\), \(F\subsetneq LF^h=(LF)^i=(LF)^r=LF\) (so \(F\subsetneq L^iF=(LF)^d=LF\)). They also investigate examples where \(F\) is a rational function field.
Reviewer: Gerard Leloup (Le Mans)Study of the algebra of smooth integro-differential operators with applications.https://www.zbmath.org/1456.160212021-04-16T16:22:00+00:00"Haghany, A."https://www.zbmath.org/authors/?q=ai:haghany.ahmad"Kassaian, Adel"https://www.zbmath.org/authors/?q=ai:kassaian.adelBivariate Kolchin-type dimension polynomials of non-reflexive prime difference-differential ideals. The case of one translation.https://www.zbmath.org/1456.120042021-04-16T16:22:00+00:00"Levin, Alexander"https://www.zbmath.org/authors/?q=ai:levin.alexander.1|levin.alexander|levin.alexander-bSee the combined review about four papers by Alexander Levin (one coauthored with Alexander Egrafov) in Zbl 1456.12002.
\begin{itemize}
\item I: \textit{A. Levin}, Math. Comput. Sci. 13, No. 1--2, 157--168 (2019);
\item II: \textit{A. Levin}, J. Symb. Comput. 102, 173--188 (2021);
\item III: \textit{A. Levin}, Math. Comput. Sci. 14, No. 2, 361--374 (2020; Zbl 1456.12001);
\item IV: \textit{A. Evgrafov} and \textit{A. Levin}, Math. Comput. Sci. 14, No. 2, 347--360 (2020; Zbl 1456.12003).
\end{itemize}
Reviewer: Jacques Sauloy (Toulouse)Bivariate dimension quasi-polynomials of difference-differential field extensions with weighted basic operators.https://www.zbmath.org/1456.120022021-04-16T16:22:00+00:00"Levin, Alexander"https://www.zbmath.org/authors/?q=ai:levin.alexander-bThe present review is about the four following papers by Alexander Levin, one coauthored with Alexander Egrafov:
\begin{itemize}
\item I: \textit{A. Levin}, Math. Comput. Sci. 13, No. 1--2, 157--168 (2019);
\item II: \textit{A. Levin}, J. Symb. Comput. 102, 173--188 (2021; Zbl 1456.12004);
\item III: \textit{A. Levin}, Math. Comput. Sci. 14, No. 2, 361--374 (2020; Zbl 1456.12003)
\item IV: \textit{A. Evgrafov} and \textit{A. Levin}, Math. Comput. Sci. 14, No. 2, 347--360 (2020; Zbl 1456.12001).
\end{itemize}
We shall also have to refer to some more ancient works:
\begin{itemize}
\item \textit{A. Einstein}, The meaning of relativity. Fourth ed. Princeton: Princeton University Press (1953; Zbl 0050.21208)
\item \textit{E. R. Kolchin}, Bull. Am. Math. Soc. 70, 570--573 (1964; Zbl 0144.03702)
\item \textit{M. V. Kondratieva} et al., Differential and difference dimension polynomials. Dordrecht: Kluwer Academic Publishers (1999; Zbl 0930.12005)
\item \textit{A. B. Levin}, Difference algebra. New York, NY: Springer (2008; Zbl 1209.12003).
\end{itemize}
The four papers under review are related to systems of difference, resp. difference-differential equations; and, more precisely, to the number of arbitrary parameters in a generic solution of such systems. Differential algebra evolved from commutative algebra, especially in its algebraic geometry component; and difference algebra evolved from differential algebra. We briefly recall the history of the relevant concepts.
The number of arbitrary parameters in a generic solution of a system of algebraic equations in \(m\) variables is the \textit{dimension} of the corresponding algebraic subvariety of the affine space of dimension \(m\). This dimension, as well as other important geometric invariants (such as the multiplicity of a solution) are encoded in the \textit{Hilbert polynomial}, a powerful tool rooted in commutative algebra, but resting in a linear point of view as follows: for some graded module \(\bigoplus M_n\), the function \(n \mapsto \dim_K M_n\) (\(K\) the base field) is polynomial for \(n \gg 0\) (meaning \(n\) ``big enough''). See the book of \textit{D. Eisenbud}, ``Commutative algebra. With a view toward algebraic geometry.'' Berlin: Springer-Verlag (1995; Zbl 0819.13001) for details.
The corresponding concepts for systems of differential equations were introduced and developed by Kolchin in 1964: the \textit{differential dimension} is ``the number of arbitrary functions [...] in the solution of the system [...]'' [Kolchin (loc. cit.)]. The \textit{differential dimension polynomial} also appears there, in order to replace the Hilbert polynomial. Note that a more down-to-earth, yet maybe more detailed notion had been previously introduced by Einstein in 1953 under the name of \textit{strength} of a system of partial differential equations [Einstein (loc. cit.)]. For a precise definition of the \textit{Einstein strength}, see [Kondratieva et al., (loc. cit.)] and [Levin I]. The former book develops Kolchin theory and builds a difference analogue; the latter lies entirely within the frame of difference algebra.
The four papers under review here deal with the computation of dimension polynomials. The first three involve systems of difference-differential equations, while the fourth involves ``pure'' difference equations. As is the case for Hilbert polynomials in commutative algebra and algebraic geometry, the techniques required include: theory of polynomials taking only integer values; some combinatorics of lattices (used to model the properties of multi-index exponents, weighted or not, of multi-variate polynomials); and a theory of reduction of polynomials (compare e.g. with Gröbner bases) in which a key ingredient is the construction of \textit{characteristic sets}
[\textit{M. V. Kondratieva} et al., (loc. cit.); Levin I].
We now describe more precisely the framework of the first article [ Levin I]. Then we shall explain the variations for the three other ones.
A \textit{difference-differential field} is a field \(K\) endowed with a set \(\Delta := \{\delta_1,\ldots,\delta_m\}\) of derivations and a set \(\sigma := \{\sigma_1,\ldots,\sigma_n\}\) of automorphisms (Thus the setting described here is that of \textit{inversive} difference fields; but actually some parts of the theory are valid for rings, assuming only that the \(\sigma_j\) are injective endomorphisms (which is of course automatic in the case of fields).), all the \(\delta_i\) and \(\sigma_j\) commuting among themselves. More precise terminology is: \(\Delta-\sigma\)-field. Notions of \(\Delta-\sigma\)-subfield, extension field, homomorphism are easily defined; as well as those of \(\Delta-\sigma\)-polynomials (generalizing differential polynomials), independence and transcendence degree.
Write \(T := \{\delta_1^{k_1} \cdots \delta_m^{k_m} \sigma_1^{l_1} \cdots \sigma_n^{l_n}\}\), all \(k_i, l_j \in \mathbf{N}\), the free commutative semigroup on \(\Delta \cup \sigma\).
For integers \(r,s\), let \(T(r,s)\) the subset of those \(\tau := \delta_1^{k_1} \cdots \delta_m^{k_m} \sigma_1^{l_1} \cdots \sigma_n^{l_n}\) such that \(\text{ord}(\tau) \leq r\) and \(\text{ord}'(\tau) \leq s\), where \(\text{ord}(\tau) := \sum k_i\) and \(\text{ord}'(\tau) := \sum l_j\).
For any \(\Delta-\sigma\)-extension \(L\) of \(K\) generated by \(\eta_1,\ldots,\eta_p\), and any integers \(r,s\), let \(L_{r,s}\) the subextension generated by all the \(\tau \eta_1,\ldots,\tau \eta_p\), \(\tau \in T(r,s)\). Then the main theorem of [Levin I] article is:
\textit{For \(r,s >>0\), the transcendence degree of \(L_{r,s}\) is polynomial as a function
of \(r,s\).}
In [II], the global setting is the same but there is only one difference automorphism \(\sigma\). Then the above theorem is refined in the following way:
The polynomial above has degree \(1\) in \(s\); moreover an explicit formula is given for its two coefficients (polynomials in \(r\)) from some combinatorial computations.
In [Levin III], the global setting is that of [Levin I], but the sets \(\Delta\) and \(\sigma\) are respectively partitioned in families of subsets \((\Delta_i)_{i= 1,\ldots,p}\), \((\sigma_j)_{j = 1,\ldots,q}\). All weights are taken to be \(1\) and partial weights \(\text{ord}_i(\tau)\), resp. \(\text{ord}'_j(\tau)\) are defined relative to the \(\Delta_i\), resp. the \(\sigma_j\), yielding subsets (We change slightly the notations to emphasize homogeneity with [Levin I].) \(T(r_1,\ldots,r_p,s_1,\ldots,s_q)\) of \(T\) defined by conditions \(\text{ord}_i(\tau) \leq r_i\), resp. \(\text{ord}'_j(\tau) \leq s_j\). Subextensions \(L(r_1,\ldots,r_p,s_1,\ldots,s_q)\) of \(L\) are defined accordingly. Then:
\textit{The transcendence degree of \(L(r_1,\ldots,r_p,s_1,\ldots,s_q)\) is polynomial as a function of the \(r_i,s_j\) (for values \(>> 0\)).}
Finally, in [Levin IV], \(\Delta\) is empty (no derivation, ``pure difference'' case). But then the corresponding dimension polynomial is related to Einstein's strength and application is given to equation arising from natural sciences.
Reviewer: Jacques Sauloy (Toulouse)Workbook algebra. Problems and solutions with detailed explanations and hints. 2nd edition.https://www.zbmath.org/1456.000012021-04-16T16:22:00+00:00"Karpfinger, Christian"https://www.zbmath.org/authors/?q=ai:karpfinger.christianPublisher's description: Dieses Buch erleichtert Ihnen den Einstieg in das eigenständige Lösen von Aufgaben zur Algebra, indem es Ihnen nicht einfach nur Aufgaben mit Lösungen, sondern vor allem auch Hinweise zur Lösungsfindung und ausführliche Motivationen bietet.
Damit ist das Werk ideal geeignet zur Prüfungsvorbereitung, wenn Sie ein tieferes Verständnis der Algebra entwickeln wollen oder wenn Sie sich gerne an kniffligen Aufgaben einer faszinierenden mathematischen Disziplin versuchen.
In den mehr als 300 Aufgaben unterschiedlicher Schwierigkeitsgrade durchleuchten wir die grundlegenden algebraischen Strukturen Gruppen, Ringe und Körper, wie sie typischerweise in einer Anfängervorlesung für Mathematikstudierende behandelt werden.
Vielfach berufen wir uns in den Lösungen auf Sätze, Lemmata und Korollare der 5. Auflage des Buches [\textit{C. Karpfinger} and \textit{K. Meyberg}, Algebra. Gruppen, Ringe, Körper. 5th edition. Berlin: Springer Spektrum (2021; Zbl 07301963)]
See the review of the first edition in [Zbl 1309.00004]. For the previous editions of the textbook see [Zbl 1181.00002; Zbl 1210.00005; Zbl 1259.00001; Zbl 1365.00002].On twists of smooth plane curves.https://www.zbmath.org/1456.111172021-04-16T16:22:00+00:00"Badr, Eslam"https://www.zbmath.org/authors/?q=ai:badr.eslam-e"Bars, Francesc"https://www.zbmath.org/authors/?q=ai:bars.francesc"Lorenzo García, Elisa"https://www.zbmath.org/authors/?q=ai:lorenzo-garcia.elisaLet \(C\) be a projective, smooth, non-hyperelliptic, curve and genus \(g \geq 3\) defined over a field \(k\). Denote by \(\bar{k }\) a fixed separable closure of \(k\) and by \(\bar{C}\) the curve \(C \times_k \bar{k}\). A twist of \(C\) over \(k\) is a projective, non-singular \(C^{\prime}\) defined over \(k\) with a \(\bar{k}\)-isomorphism \(\varphi : \overline{C^{\prime}} \rightarrow \bar{C}\). The paper under review deals with the following question: Assuming that \(C\) admits a smooth \(\bar{k}\)-plane model, does it have a smooth plane model over \(k\)? And if the answer is yes, does every twist of C over
\(k\) also have smooth plane model over \(k\)? The answer, in general, is negative. The twists possessing such models are characterized and an example of a twist not admitting any non-singular plane model over \(k\) is given. An interesting consequence is that explicit equations for a non-trivial Brauer-Severi surface are obtained. Furthermore, for smooth plane curves defined over \(k\) with a cyclic automorphism group generated by a diagonal matrix, a general theoretical result to
compute all its twists is presented.
Reviewer: Dimitros Poulakis (Thessaloniki)On Brauer \(p\)-dimensions and absolute Brauer \(p\)-dimensions of Henselian fields.https://www.zbmath.org/1456.160152021-04-16T16:22:00+00:00"Chipchakov, Ivan D."https://www.zbmath.org/authors/?q=ai:chipchakov.ivan-dLet \(E\) be a field. For each prime number \(p\), let \(Brd_p(E)\) be the Brauer \(p\)-dimension of \(E\) and \(abrd_p(E)\) be the absolute Brauer \(p\)-dimension of \(E\). In this paper, the author studies the sequences \(Brd_p(E)\) and \(abrd_p(E)\), when \(p\) runs over the set \(\mathbb{P}\) of the prime numbers. Suppose that \((E,v)\) is a Henselian valued field with residue field \(\hat E\). For \(p\in\mathbb{P}\), under some restrictions on \(\hat E\), such as \(abrd_p(\hat E)=0\), the author determines
\(Brd_p(E)\) and \(abrd_p(E)\) if \(\mathrm{char}(\hat E)\not=p\). Let \(\Sigma_0\) be the set of sequences \(Brd_p(E)\) and \(abrd_p(E)\) where \(p\in \mathbb{P}\) and \(E\) runs across the class of Henselian valued fields with \(\mathrm{char}(\hat E)\not=0\) and a projective absolute Galois group \(\mathcal G_{\hat E}\). The author gives a description for \(\Sigma_0\). Especially, \(\Sigma_0\) admits a sequence \(a_p,b_p\in\mathbb{N}\cup\{0,\infty\}\), \(p\in \mathbb{P}\), with \(a_2\leq 2b_2\) and
\(a_p\geq b_p\). The paper covers similar results in the case of nonzero characteristic.
Reviewer: Ali Benhissi (Monastir)On a type of permutation rational functions over finite fields.https://www.zbmath.org/1456.112192021-04-16T16:22:00+00:00"Hou, Xiang-dong"https://www.zbmath.org/authors/?q=ai:hou.xiang-dong"Sze, Christopher"https://www.zbmath.org/authors/?q=ai:sze.christopherSummary: Let \(p\) be a prime and \(n\) be a positive integer. Let \(f_b(X)=X+(X^p-X+b)^{-1}\), where \(b\in\mathbb{F}_{p^n}\) is such that \(\text{Tr}_{p^n/p}(b)\neq 0\). In 2008, \textit{J. Yuan} et al. [Finite Fields Appl. 14, No. 2, 482--493 (2008; Zbl 1211.11136)] showed that for \(p=2,3,f_b\) permutes \(\mathbb{F}_{p^n}\) for all \(n\geq 1\). Using the Hasse-Weil bound, we show that when \(p>3\) and \(n\geq 5, f_b\) does not permute \(\mathbb{F}_{p^n} \). For \(p>3\) and \(n=2\), we prove that \(f_b\) permutes \(\mathbb{F}_{p^2}\) if and only if \(\text{Tr}_{p^2/p}(b)=\pm 1\). We conjecture that for \(p>3\) and \(n=3,4,f_b\) does not permute \(\mathbb{F}_{p^n}\).