相关论文: Short Rational Functions for Toric Algebra and App…

Hadamard ideals were introduced in 2006 as a set of nonlinear polynomial equations whose zeros are uniquely related to Hadamard matrices with one or two circulant cores of a given order. Based on this idea, the cocyclic Hadamard test enable…

We characterize the entire functions $P$ of $d$ variables, $d\ge 2,$ for which the $\mzd$-translates of $P\chi_{[0,N]^d}$ satisfy the partition of unity for some $N\in \mn.$ In contrast to the one-dimensional case, these entire functions…

A toric ideal is called robust if its universal Gr\"obner basis is a minimal set of generators, and is called generalized robust if its universal Gr\"obner basis equals its universal Markov basis (the union of all its minimal sets of…

Given a $d \times n$ integer matrix $A$, the main result is an elementary, simple-to-state algorithm that finds the largest $A$-graded ideal contained in any ideal $I$ in a polynomial ring $\Bbbk[x_1,\ldots,x_n]$. The special case where $A$…

We propose a numerical linear algebra based method to find the multiplication operators of the quotient ring $\mathbb{C}[x]/I$ associated to a zero-dimensional ideal $I$ generated by $n$ $\mathbb{C}$-polynomials in $n$ variables. We assume…

This article discusses a computational treatment of the localization A_L of an affine coordinate ring A at a prime ideal L and its associated graded ring Gr_a(A_L) with the means of standard basis techniques. Building on Mora's work, we…

In this expositional paper, we discuss commutative algebra -- a study inspired by the properties of integers, rational numbers, and real numbers. In particular, we investigate rings and ideals, and their various properties. After, we…

For a finite group $G$, Frobenius found a formula for the values of the function $\sum_{\mathrm{Irr} G} (\dim\, \pi)^{-s}$ for even integers $s$, where $\mathrm{Irr} G$ is the set of irreducible representations of $G$. We generalize this…

An algorithm to generate a minimal comprehensive Gr\"obner\, basis of a parametric polynomial system from an arbitrary faithful comprehensive Gr\"obner\, system is presented. A basis of a parametric polynomial ideal is a comprehensive…

Partition of unities appear in many places in analysis. Typically they are generated by compactly supported functions with a certain regularity. In this paper we consider partition of unities obtained as integer-translates of entire…

$\DeclareMathOperator{\IntR}{Int{}^\text{R}}$Integer-valued rational functions are a natural generalization of integer-valued polynomials. Given a domain $D$, the collection of all integer-valued rational functions over $D$ forms a ring…

Let $q$ be an odd prime power and let $X(m,q)$ be the set of symmetric bilinear forms on an $m$-dimensional vector space over $\mathbb{F}_q$. The partition of $X(m,q)$ induced by the action of the general linear group gives rise to a…

This is a survey article featuring some of Wolmer Vasconcelos' contributions to commutative algebra, and explaining how Vasconcelos' work and insights have contributed to the development of commutative algebra and its interaction with other…

A \textit{symmetric ideal} $I \subseteq R = K[x_1,x_2,...]$ is an ideal that is invariant under the natural action of the infinite symmetric group. We give an explicit algorithm to find Gr\"obner bases for symmetric ideals in the infinite…

Let $I = ( f_1, \dots, f_n )$ be a homogeneous ideal in the polynomial ring $K[x_1, \dots,x_n]$ over a field $K$ generated by generic polynomials. Using an incremental approach based on a method by Gao, Guan and Volny, and properties of the…

Let $(R,\mathfrak m)$ be an analytically unramified local ring of positive prime characteristic $p.$ For an ideal $I$, let $I^*$ denote its tight closure. We introduce the tight Hilbert function $H^*_I(n)=\ell(R/(I^n)^*)$ and the…

The toric ring together with the toric ideal arising from a nested configuration is studied, with particular attention given to the algebraic study of normality of the toric ring as well as the Gr\"obner bases of the toric ideal. One of the…

In this paper we present a new efficient variant to compute strong Gr\"obner basis over quotients of principal ideal domains. We show an easy lifting process which allows us to reduce one computation over the quotient $R/nR$ to two…

Associated to any hypergraph is a toric ideal encoding the algebraic relations among its edges. We study these ideals and the combinatorics of their minimal generators, and derive general degree bounds for both uniform and non-uniform…

In this paper we describe the equations defining the multi-Rees algebra $k[x_1,\dots,x_n][I_1^{a_1}t_1,\dots,I_r^{a_r}t_r]$, where the ideals are generated by subsets of $x_1,\dots,x_n$. We also show that a family of binomials whose leading…