A note on the stability of pencils of plane curves

We investigate the problem of classifying pencils of plane curves of degree d up to projective equivalence. We obtain explicit stability criteria in terms of the log canonical threshold by relating the stability of a pencil to the stability of the curves lying on it.

The study of the projective transformation under the bilinear strict equivalence

The study of linear time invariant descriptor systems has intimately been related to the study of matrix pencils. It is true that a large number of systems can be reduced to the study of differential (or difference) systems, $S\left ( {F,G} \right )$, $$\begin{align*} & S\left({F,G}\right): F\dot{x}(t) = G{x}(t) \left(\text{or the dual, } F{x}(t) = G\dot{x}(t)\right), \end{align*}$$and $$\begin{align*} & S\left({F,G}\right): Fx_{k+1} = Gx_k \left(\text{or the dual, } Fx_k=Gx_{k+1}\right)\!, F,G \in{\mathbb{C}^{m \times n}}, \end{align*}$$and their properties can be characterized by homogeneous matrix pencils, $sF - \hat{s}G$. Based on the fact that the study of the invariants for the projective equivalence class can be reduced to the study of the invariants of the matrices of set ${\mathbb{C}^{k \times 2}}$ (for $k \geqslant 3$ with all $2\times 2$-minors non-zero) under the extended Hermite equivalence, in the context of the bilinear strict equivalence relation, a novel projective transformation is analytically derived.

Sign Variation and Descents

For any $n > 0$ and $0 \leq m < n$, let $P_{n,m}$ be the poset of projective equivalence classes of $\{-,0,+\}$-vectors of length $n$ with sign variation bounded by $m$, ordered by reverse inclusion of the positions of zeros. Let $\Delta_{n,m}$ be the order complex of $P_{n,m}$. A previous result from the third author shows that $\Delta_{n,m}$ is Cohen-Macaulay over $\mathbb{Q}$ whenever $m$ is even or $m = n-1$. Hence, it follows that the $h$-vector of $\Delta_{n,m}$ consists of nonnegative entries. Our main result states that $\Delta_{n,m}$ is partitionable and we give an interpretation of the $h$-vector when $m$ is even or $m = n-1$. When $m = n-1$ the entries of the $h$-vector turn out to be the new Eulerian numbers of type $D$ studied by Borowiec and Młotkowski in [ Electron. J. Combin., 23(1):#P1.38, 2016]. We then combine our main result with Klee's generalized Dehn-Sommerville relations to give a geometric proof of some facts about these Eulerian numbers of type $D$.

Supertropical quadratic forms II: Tropical trigonometry and applications

This paper is a sequel of [Z. Izhakian, M. Knebusch and L. Rowen, Supertropical semirings and supervaluations, J. Pure Appl. Algebra 220(1) (2016) 61–93], where we introduced quadratic forms on a module [Formula: see text] over a supertropical semiring [Formula: see text] and analyzed the set of bilinear companions of a quadratic form [Formula: see text] in case the module [Formula: see text] is free, with fairly complete results if [Formula: see text] is a supersemifield. Given such a companion [Formula: see text], we now classify the pairs of vectors in [Formula: see text] in terms of [Formula: see text] This amounts to a kind of tropical trigonometry with a sharp distinction between the cases for which a sort of Cauchy–Schwarz (CS) inequality holds or fails. This distinction is governed by the so-called CS-ratio [Formula: see text] of a pair of anisotropic vectors [Formula: see text] in [Formula: see text]. We apply this to study the supertropicalizations (cf. [Z. Izhakian, M. Knebusch and L. Rowen, Supertropical semirings and supervaluations, J. Pure Appl. Algebra 220(1) (2016) 61–93]) of a quadratic form on a free module [Formula: see text] over a field in the simplest cases of interest where [Formula: see text]. In the last part of the paper, we introduce a suitable equivalence relation on [Formula: see text], whose classes we call rays. (It is coarser than usual projective equivalence.) For anisotropic [Formula: see text] the CS-ratio [Formula: see text] depends only on the rays of [Formula: see text] and [Formula: see text]. We develop essential basics for a kind of convex geometry on the ray-space of [Formula: see text], where the CS-ratios play a major role.

Bounds related to projective equivalence classes of good filtrations

Let [Formula: see text] be a regular ideal in noetherian ring [Formula: see text]. Mc Adam and Ratliff showed the existence of the unique minimal reduction number of [Formula: see text], noted [Formula: see text], such that for every minimal reduction [Formula: see text] of [Formula: see text], [Formula: see text] and [Formula: see text]. They showed that the set of integers [Formula: see text] is bounded in terms of the analytic spread of [Formula: see text]. Here, we extend these results to good filtrations. Let [Formula: see text] be a good filtration on [Formula: see text], we show that the set of integers [Formula: see text] is bounded.

On KM-Arcs in Small Desarguesian Planes

In this paper we study the existence problem for KM-arcs in small Desarguesian planes. We establish a full classification of KM$_{q,t}$-arcs for $q\le 32$, up to projective equivalence. We also construct a KM$_{64,4}$-arc; as $t=4$ was the only value for which the existence of a KM$_{64,t}$-arc was unknown, this fully settles the existence problem for $q\le 64$.

On Fε2-planar mappings with function ε of (pseudo-)Riemannian manifolds

In this paper we study special mappings between n-dimensional (pseudo-) Riemannian manifolds. In 2003 Topalov introduced PQ?-projectivity of Riemannian metrics, with constant ? ? 0,1 + n. These mappings were studied later by Matveev and Rosemann and they found that for ? = 0 they are projective. These mappings could be generalized for case, when ? will be a function on manifold. We show that PQ?- projective equivalence with ? is a function corresponds to a special case of F-planar mapping, studied by Mikes and Sinyukov (1983) with F = Q. Moreover, the tensor P is derived from the tensor Q and non-zero function ?. We assume that studied mappings will be also F2-planar (Mikes 1994). This is the reason, why we suggest to rename PQ? mapping as F?2. For these mappings we find the fundamental partial differential equations in closed linear Cauchy type form and we obtain new results for initial conditions.

Classification of Elliptic Cubic Curves Over The Finite Field of Order Nineteen

Plane cubics curves may be classified up to isomorphism or projective equivalence. In this paper, the inequivalent elliptic cubic curves which are non-singular plane cubic curves have been classified projectively over the finite field of order nineteen, and determined if they are complete or incomplete as arcs of degree three. Also, the maximum size of a complete elliptic curve that can be constructed from each incomplete elliptic curve are given.

A note on Gorenstein transposes

Let [Formula: see text] be a left and right Noetherian ring. In this paper, we prove that any Gorenstein transpose of a finitely generated [Formula: see text]-module is exactly an Auslander transpose. As applications, we obtain a new relation between a Gorenstein transpose of a module with a transpose of the same module, and show that the Gorenstein transpose of a module is unique up to Gorenstein projective equivalence. In addition, when [Formula: see text] is an Artin algebra, the corresponding Auslander–Reiten sequences are constructed in terms of Gorenstein transposes.