Recent zbMATH articles in MSC 51E https://www.zbmath.org/atom/cc/51E 2021-04-16T16:22:00+00:00 Werkzeug Results on partial geometries with an abelian Singer group of rigid type. https://www.zbmath.org/1456.51004 2021-04-16T16:22:00+00:00 "De Winter, Stefaan" https://www.zbmath.org/authors/?q=ai:de-winter.stefaan "Kamischke, Ellen" https://www.zbmath.org/authors/?q=ai:kamischke.ellen "Neubert, Eric" https://www.zbmath.org/authors/?q=ai:neubert.eric "Wang, Zeying" https://www.zbmath.org/authors/?q=ai:wang.zeying A partial geometry $$pg(s,t,\alpha)$$ is a finite partial linear space such that every line contains $$s+1$$ points, every point lies on $$t+1$$ lines, and given a point $$P$$ not incident with a line $$L$$, there are exactly alpha lines through $$P$$ meeting $$L$$. The partial geometry is said to admit a Singer group if its automorphism group contains a subgroup acting sharply transitively on the points. If the Singer group is an abelian group G, then it was shown in [\textit{S. De Winter}, J. Algebr. Comb. 24, No. 3, 285--297 (2006; Zbl 1106.51003)] that the stabiliser of a line in G is either trivial or has size $$s+1$$. The authors call the partial geometry, together with the group G, of rigid type'' if all lines have a trivial stabiliser. The only known example of a partial geometry of rigid type is due to Van Lint and Schrijver and has parameters $$(5,5,2)$$. In this paper, the authors study the possible parameters of partial geometries of rigid type (and their associated Singer group). By use of different necessary conditions on the parameters and some properties of the associated group, and aided by a computer, the authors show that if $$\alpha\leq 1000$$, there are only five parameter sets that could possibly correspond to a partial geometry of rigid type. The hypothetical partial geometries would have one of the four sets of parameters $$(11,23,3), (39,39,15), (272,272,104)$$ or $$(2295,4591,615)$$. The authors resist the temptation to conjecture that the Van Lint-Schrijver partial geometry is the only partial geometry of rigid type. Instead they formulate the weaker conjecture that for a partial geometry of rigid type, the number of points is either a power of $$2$$ or a power of $$3$$. Note that this conjecture does not rule out the existence of a partial geometry of rigid type with parameters $$(11,23,3)$$ or $$(39,39,15)$$. Reviewer: Geertrui Van de Voorde (Canterbury) 50 years of finite geometry, the geometries over finite rings'' part. https://www.zbmath.org/1456.51002 2021-04-16T16:22:00+00:00 "Keppens, Dirk" https://www.zbmath.org/authors/?q=ai:keppens.dirk Summary: Whereas for a substantial part, finite geometry'' during the past 50 years has focussed on geometries over finite fields, geometries over finite rings that are not division rings have got less attention. Nevertheless, several important classes of finite rings give rise to interesting geometries. In this paper we bring together some results, scattered over the literature, concerning finite rings and plane projective geometry over such rings. The paper does not contain new material, but by collecting information in one place, we hope to stimulate further research in this area for at least another 50 years of finite geometry. On linear sets of minimum size. https://www.zbmath.org/1456.51006 2021-04-16T16:22:00+00:00 "Jena, Dibyayoti" https://www.zbmath.org/authors/?q=ai:jena.dibyayoti "Van de Voorde, Geertrui" https://www.zbmath.org/authors/?q=ai:van-de-voorde.geertrui Summary: An $$\mathbb{F}_q$$-linear set of rank $$k$$, $$k \leq h$$, on a projective line $$\mathrm{PG} ( 1 , q^h )$$, containing at least one point of weight one, has size at least $$q^{k - 1} + 1$$ (see [\textit{J. De Beule} and the second author, J. Comb. Theory, Ser. A 164, 109--124 (2019; Zbl 1411.51004)]). The classical example of such a set is given by a club. In this paper, we construct a broad family of linear sets meeting this lower bound, where we are able to prescribe the weight of the heaviest point to any value between $$k \slash 2$$ and $$k - 1$$. Our construction extends the known examples of linear sets of size $$q^{k - 1} + 1$$ in $$\mathrm{PG} ( 1 , q^h )$$ constructed for $$k = h = 4$$ [\textit{G. Bonoli} and \textit{O. Polverino}, Innov. Incidence Geom. 2, 35--56 (2005; Zbl 1103.51006)] and $$k = h$$ in [\textit{G. Lunardon} and \textit{O. Polverino}, J. Comb. Theory, Ser. A 90, No. 1, 148--158 (2000; Zbl 0964.51009)]. We determine the weight distribution of the constructed linear sets and describe them as the projection of a subgeometry. For small $$k$$, we investigate whether all linear sets of size $$q^{k - 1} + 1$$ arise from our construction. Finally, we modify our construction to define rank $$k$$ linear sets of size $$q^{k - 1} + q^{k - 2} + \dots + q^{k - l} + 1$$ in $$\mathrm{PG} ( l , q^h )$$. This leads to new infinite families of small minimal blocking sets which are not of Rédei type. The homogeneous pseudo-embeddings and hyperovals of the generalized quadrangle $$H(3,4)$$. https://www.zbmath.org/1456.51003 2021-04-16T16:22:00+00:00 "De Bruyn, Bart" https://www.zbmath.org/authors/?q=ai:de-bruyn.bart "Gao, Mou" https://www.zbmath.org/authors/?q=ai:gao.mou Let $$\mathcal{S}=(\mathcal{P},\mathcal{L})$$ be a generalized quadrangle and denote by $$\varepsilon:\mathcal{S} \longrightarrow \mathrm{PG}(V)$$ the universal pseudo-embedding $$\mathcal{S}.$$ The vector dimension of the subspace $$\langle \varepsilon(\mathcal{P})\rangle$$ of $$V$$ is called the \textit{pseudo-embedding rank} of $$\mathcal{S}.$$ If $$|\mathcal{P}|$$ is finite, denote by $$C=C(\mathcal{S})$$ the binary code of length $$|\mathcal{P}|$$ generated by the characteristic vectors of the lines of $$\mathcal{S}.$$ It has been proved in [the first author, Adv. Geom. 13, No. 1, 71--95 (2013; Zbl 1267.51002)] that the pseudo-embedding rank of $$\mathcal{S}$$ is $$|\mathcal{P}|- \dim(C).$$ In this paper, the authors prove that the pseudo-embedding rank of the Hermitian quadrangle $$H(3,4)$$ is equal to $$24.$$ As a consequence, the binary code $$C(H(3,4))$$ has dimension $$45-24=21,$$ because the generalized quadrangle $$H(3,4)$$ has $$45$$ points. They also show that there are, up to isomorphism, four homogeneous pseudo-embeddings of $$H(3,4),$$ with respective vector dimensions $$14,$$ $$15,$$ $$23$$ and $$24.$$ Reviewer: Guglielmo Lunardon (Napoli) On the balanced upper chromatic number of finite projective planes. https://www.zbmath.org/1456.05054 2021-04-16T16:22:00+00:00 "Blázsik, Zoltán L." https://www.zbmath.org/authors/?q=ai:blazsik.zoltan-l "Blokhuis, Aart" https://www.zbmath.org/authors/?q=ai:blokhuis.aart "Miklavič, Štefko" https://www.zbmath.org/authors/?q=ai:miklavic.stefko "Nagy, Zoltán Lóránt" https://www.zbmath.org/authors/?q=ai:nagy.zoltan-lorant "Szőnyi, Tamás" https://www.zbmath.org/authors/?q=ai:szonyi.tamas Summary: In this paper, we study vertex colorings of hypergraphs in which all color class sizes differ by at most one (balanced colorings) and each hyperedge contains at least two vertices of the same color (rainbow-free colorings). For any hypergraph $$H$$, the maximum number $$k$$ for which there is a balanced rainbow-free $$k$$-coloring of $$H$$ is called the balanced upper chromatic number of the hypergraph. We confirm the conjecture of \textit{G. Araujo-Pardo} et al. [ibid. 338, No. 12, 2562--2571 (2015; Zbl 1317.05046)] by determining the balanced upper chromatic number of the desarguesian projective plane $$\operatorname{PG} ( 2 , q )$$ for all $$q$$. In addition, we determine asymptotically the balanced upper chromatic number of several families of non-desarguesian projective planes and also provide a general lower bound for arbitrary projective planes using probabilistic methods which determines the parameter up to a multiplicative constant. Combinatorial structures in algebra and geometry. NSA 26, Constanţa, Romania, August 26 -- September 1, 2018. https://www.zbmath.org/1456.05001 2021-04-16T16:22:00+00:00 "Stamate, Dumitru I. (ed.)" https://www.zbmath.org/authors/?q=ai:stamate.dumitru-ioan "Szemberg, Tomasz (ed.)" https://www.zbmath.org/authors/?q=ai:szemberg.tomasz Publisher's description: This proceedings volume presents selected, peer-reviewed contributions from the 26th National School on Algebra, which was held in Constanţa, Romania, on August 26-September 1, 2018. The works cover three fields of mathematics: algebra, geometry and discrete mathematics, discussing the latest developments in the theory of monomial ideals, algebras of graphs and local positivity of line bundles. Whereas interactions between algebra and geometry go back at least to Hilbert, the ties to combinatorics are much more recent and are subject of immense interest at the forefront of contemporary mathematics research. Transplanting methods between different branches of mathematics has proved very fruitful in the past -- for example, the application of fixed point theorems in topology to solving nonlinear differential equations in analysis. Similarly, combinatorial structures, e.g., Newton-Okounkov bodies, have led to significant advances in our understanding of the asymptotic properties of line bundles in geometry and multiplier ideals in algebra.This book is intended for advanced graduate students, young scientists and established researchers with an interest in the overlaps between different fields of mathematics. A volume for the 24th edition of this conference was previously published with Springer under the title Multigraded Algebra and Applications'' [\textit{V. Ene} (ed.) and \textit{E. Miller} (ed.), Multigraded algebra and applications. NSA 24, Moieciu de Sus, Romania, August 17--24, 2016. Cham: Springer (2018; Zbl 1400.13003)]. The articles of this volume will be reviewed individually. Indexed articles: \textit{Andrei-Ciobanu, Claudia}, Nearly normally torsionfree ideals, 1-13 [Zbl 1453.13007] \textit{Cimpoeaş, Mircea; Stamate, Dumitru I.}, Gröbner-nice pairs of ideals, 15-29 [Zbl 1453.13077] \textit{Dumnicki, Marcin; Farnik, Łucja; Harbourne, Brian; Szemberg, Tomasz; Tutaj-Gasińska, Halszka}, Veneroni maps, 31-42 [Zbl 1453.14043] \textit{Ene, Viviana; Herzog, Jürgen}, On the symbolic powers of binomial edge ideals, 43-50 [Zbl 1456.05183] \textit{Erey, Nursel}, Multigraded Betti numbers of some path ideals, 51-65 [Zbl 1456.05075] \textit{Hibi, Takayuki; Tsuchiya, Akiyoshi}, Depth of an initial ideal, 67-71 [Zbl 1453.13081] \textit{Juhnke-Kubitzke, Martina; Le, Dinh Van; Römer, Tim}, Asymptotic behavior of symmetric ideals: a brief survey, 73-94 [Zbl 1453.13024] \textit{Pegel, Christoph; Sanyal, Raman}, On piecewise-linear homeomorphisms between distributive and anti-blocking polyhedra, 95-114 [Zbl 1457.52014] \textit{Popescu, Dorin}, The Bass-Quillen conjecture and Swan's question, 115-121 [Zbl 1453.13026] \textit{Rinaldo, Giancarlo; Terai, Naoki; Yoshida, Ken-Ichi}, Licci level Stanley-Reisner ideals with height three and with type two, 123-142 [Zbl 1453.13062] \textit{Seyed Fakhari, S. A.}, Homological and combinatorial properties of powers of cover ideals of graphs, 143-159 [Zbl 1453.13036] \textit{Szpond, Justyna}, Fermat-type arrangements, 161-182 [Zbl 1453.14022] Non-elliptic webs and convex sets in the affine building. https://www.zbmath.org/1456.05173 2021-04-16T16:22:00+00:00 "Akhmejanov, Tair" https://www.zbmath.org/authors/?q=ai:akhmejanov.tair Summary: We describe the $$\mathfrak{sl}_3$$ non-elliptic webs in terms of convex sets in the affine building. Kuperberg defined the non-elliptic web basis in his work on rank-$$2$$ spider categories. \textit{B. Fontaine} et al. [Compos. Math. 149, No. 11, 1871--1912 (2013; Zbl 1304.22016)] showed that the $$\mathfrak{s}l_3$$ non-elliptic webs are dual to CAT(0) triangulated diskoids in the affine building. We show that each such triangulated diskoid is the intersection of the min-convex and max-convex hulls of a generic polygon in the building. Choosing a generic polygon from each of the components of the Satake fiber produces (the duals of) the non-elliptic web basis. The convex hulls in the affine building were first introduced by \textit{G. Faltings} [Prog. Math. 195, 157--184 (2001; Zbl 1028.14002)] and are related to tropical convexity, as discussed in work by \textit{M. Joswig} et al. [Albanian J. Math. 1, No. 4, 187--211 (2007; Zbl 1133.52003)] and by \textit{L. Zhang} [Computing convex hulls in the affine building of $$\mathfrak{sl}_d$$'', Preprint, \url{arXiv:1811.08884}]. Joints formed by lines and a $$k$$-plane, and a discrete estimate of Kakeya type. https://www.zbmath.org/1456.52026 2021-04-16T16:22:00+00:00 "Carbery, Anthony" https://www.zbmath.org/authors/?q=ai:carbery.anthony "Iliopoulou, Marina" https://www.zbmath.org/authors/?q=ai:iliopoulou.marina Summary: Let $$\mathcal{L}$$ be a family of lines and let $$\mathcal{P}$$ be a family of $$k$$-planes in $$\mathbb{F}^n$$ where $$\mathbb{F}$$ is a field. In our first result we show that the number of joints formed by a $$k$$-plane in $$\mathcal{P}$$ together with $$(n-k)$$ lines in $$\mathcal{L}$$ is $$O_n(|\mathcal{L}| |\mathcal{P}|^{1/(n-k)})$$. This is the first sharp result for joints involving higher-dimensional affine subspaces, and it holds in the setting of arbitrary fields $$F$$. In contrast, for our second result, we work in the three-dimensional Euclidean space $$\mathbb{R}^3$$, and we establish the Kakeya-type estimate $\sum\limits_{x\in J}\left(\sum\limits_{\ell\in\mathcal{L}}\chi_\ell(x)\right)^{3/2}\lesssim|\mathcal{L}|^{3/2}$ where $$J$$ is the set of joints formed by $$\mathcal{L}$$; such an estimate fails in the setting of arbitrary fields. This result strengthens the known estimates for joints, including those counting multiplicities. Additionally, our techniques yield significant structural information on quasi-extremisers for this inequality. A Poncelet criterion for special pairs of conics in $$\mathrm{PG}(2,p^m)$$. https://www.zbmath.org/1456.51005 2021-04-16T16:22:00+00:00 "Hungerbühler, Norbert" https://www.zbmath.org/authors/?q=ai:hungerbuhler.norbert "Kusejko, Katharina" https://www.zbmath.org/authors/?q=ai:kusejko.katharina Summary: We study Poncelet's Theorem in finite projective planes over the field $$\mathrm{GF}(q), q = p^m$$ for $$p$$ an odd prime and $$m > 0$$, for a particular pencil of conics. We investigate whether we can find polygons with $$n$$ sides which are inscribed in one conic and circumscribed around the other, so-called Poncelet Polygons. By using suitable elements of the dihedral group for these pairs, we prove that the length $$n$$ of such Poncelet Polygons is independent of the starting point. In this sense Poncelet's Theorem is valid. By using Euler's divisor sum formula for the totient function, we can make a statement about the number of different conic pairs, which carry Poncelet Polygons of length $$n$$. Moreover, we will introduce polynomials whose zeros in $$\mathrm{GF}(q)$$ yield information about the relation of a given pair of conics. In particular, we can decide for a given integer $$n$$, whether and how we can find Poncelet $$n$$-gons for pairs of conics in the plane $$\mathrm{PG}(2,q)$$. Investigation graph isomorphism problem via entanglement entropy in strongly regular graphs. https://www.zbmath.org/1456.94033 2021-04-16T16:22:00+00:00 "Jafarizadeh, M. A." https://www.zbmath.org/authors/?q=ai:jafarizadeh.mohamad-ali|jafarizadeh.mohammad-ali "Eghbalifam, F." https://www.zbmath.org/authors/?q=ai:eghbalifam.f "Nami, S." https://www.zbmath.org/authors/?q=ai:nami.susan