Related papers: The Minimality of Determinantal Varieties

We prove that it is decidable if a finitely based permutation class contains infinitely many simple permutations, and establish an unavoidable substructure result for simple permutations: every sufficiently long simple permutation contains…

A polynomial of degree $\ge 2$ with coefficients in the ring of $p$-adic numbers $\mathbb{Z}_p$ is studied as a dynamical system on $\mathbb{Z}_p$. It is proved that the dynamical behavior of such a system is totally described by its…

The nondeterministic advice complexity of the P-selective sets is known to be exactly linear. Regarding the deterministic advice complexity of the P-selective sets--i.e., the amount of Karp--Lipton advice needed for polynomial-time machines…

This paper investigates defining equations for secant varieties of the variety of reducible polynomials, which geometrically encode the notions of strength and slice rank of homogeneous polynomials. We present three main results. First, we…

We find the minimal non-trivial integer variable group determinant for any dihedral group of order less than $3.79\times 10^{47}$. We think of this as the Lind-Lehmer problem for the dihedral group. We give a complete description of the…

We study the case of a real homogeneous polynomial $P$ whose minimal real and complex decompositions in terms of powers of linear forms are different. We prove that, if the sum of the complex and the real ranks of $P$ is at most $…

Orthostochastic matrices are the entrywise squares of orthogonal matrices, and naturally arise in various contexts, including notably definite symmetric determinantal representations of real polynomials. However, defining equations for the…

Let $V$ be a finite-dimensional vector space over the complex numbers and let $G\leq \operatorname{SL}(V)$ be a finite group. We describe the class group of a minimal model (that is, $\mathbb Q$-factorial terminalization) of the linear…

We classify minimal complex surfaces of general type with $p_g=q=3$. More precisely, we show that such a surface is either the symmetric product of a curve of genus 3 or a free $\Z_2-$quotient of the product of a curve of genus 2 and a…

Say a trinomial $x^n+A x^m+B \in \Q[x]$ has reducibility type $(n_1,n_2,...,n_k)$ if there exists a factorization of the trinomial into irreducible polynomials in $\Q[x]$ of degrees $n_1$, $n_2$,...,$n_k$, ordered so that $n_1 \leq n_2 \leq…

An extended formulation of a polytope P is a polytope Q which can be projected onto P. Extended formulations of small size (i.e., number of facets) are of interest, as they allow to model corresponding optimization problems as linear…

We consider a special category of Hopf algebras, depending on parameters $\Sigma$ which possess properties similar to the category of representations of simple Lie group with highest weight $\lambda$. We connect quantum groups to minimal…

We study initial algebras of determinantal rings, defined by minors of generic matrices, with respect to their classical generic point. This approach leads to very short proofs for the structural properties of determinantal rings. Moreover,…

In this paper we investigate the question of when the determinantal ring $R$ over a field $k$ is an almost Gorenstein local/graded ring in the sense of Goto, Takahashi, and the author. As a consequence of the main result, we see that if $R$…

We describe normal forms and minimal models of Picard curves, discussing various arithmetic aspects of these. We determine all so-called special Picard curves over $\mathbb{Q}$ with good reduction outside 2 and 3, and use this to determine…

For a finite group $G$, let $\text{rdim}(G)$ denote the smallest dimension of a faithful, complex linear representation of $G$. It is clear that $\text{rdim}(H)\leq \text{rdim}(G)$ for any subgroup $H$ of $G$. We consider $G$ with the…

The small finitistic dimension $\fPD(R)$ of a ring $R$ is defined to be the supremum of projective dimensions of $R$-modules with finite projective resolutions. In this paper, we investigate the small finitistic dimensions of four types of…

Denote the alternating and symmetric groups of degree $n$ by $A_n$ and $S_n$ respectively. Consider a permutation $\sigma\in S_n$ all of whose nontrivial cycles are of the same length. We find the minimal polynomials of $\sigma$ in the…

The work considers an equivalence relation in the set of all $n\times m$ matrices with entries in the set $[p]=\{ 0,1,\ldots , p-1 \}$. In each element of the factor-set generated by this relation, we define the concept of canonical matrix,…

Given a subset $\mathcal{S}\subseteq \mathbb{F}_q[x]$ and fixed integers $n,m\in \mathbb{N}$, we study the distribution of the smallest denominator $Q\in \mathcal{S}$ for which there exists $\mathbf{P}\in \mathbb{F}_q[x]^m$ such that…