Related papers: A polynomial ideal associated to any $t$-$(v,k,\la…

In a recent paper by Harada, Seceleanu, and \c{S}ega, the Hilbert function, betti table, and graded minimal free resolution of a general principal symmetric ideal are determined when the number of variables in the polynomial ring is…

In this paper we consider an algorithmic technique more general than that proposed by Zharkov and Blinkov for the involutive analysis of polynomial ideals. It is based on a new concept of involutive monomial division which is defined for a…

Let $\Gamma$ be a $Q$-polynomial distance-regular graph with diameter at least $3$. Terwilliger (1993) implicitly showed that there exists a polynomial, say $T(\lambda)\in \mathbb{C}[\lambda]$, of degree $4$ depending only on the…

We consider a statistical problem to estimate variables (effects) that are associated with the edges of a complete bipartite graph $K_{v_1, v_2}=(V_1, V_2 \, ; E)$. Each data is obtained as a sum of selected effects, a subset of $E$. In…

In this paper, we focus on the initial degree and the vanishing of the Valabrega-Valla module of a pair of monomials ideals $J\subseteq I$ in a polynomials ring over a field $\mathbb{K}$. We prove that the initial degree of this module is…

A Gotzmann monomial ideal of the polynomial ring is a monomial ideal which is generated in one degree and which satisfies Gotzmann's persistence theorem. A subset $V$ is said to be a Gotzmann subset if the ideal generated by $V$ is a…

Let $L\subset \mathbb{Z}^n$ be a lattice and $I_L=\langle x^{\bf u}-x^{\bf v}:\ {\bf u}-{\bf v}\in L\rangle$ be the corresponding lattice ideal in $\Bbbk[x_1,\ldots, x_n]$, where $\Bbbk$ is a field. In this paper we describe minimal…

Trigonometric polynomials are usually defined on the lattice of integers.We consider the larger class of weight and root lattices with crystallographic symmetry.This article gives a new approach to minimize trigonometric polynomials, which…

A $2-(v,k,\lambda)$ directed design (or simply a $2-(v,k,\lambda)DD$) is super-simple if its underlying $2-(v,k,2\lambda)BIBD$ is super-simple, that is, any two blocks of the $BIBD$ intersect in at most two points. A $2-(v,k,\lambda)DD$ is…

We consider reasoning and minimization in systems of polynomial ordinary differential equations (ode's). The ring of multivariate polynomials is employed as a syntax for denoting system behaviours. We endow this set with a transition system…

We study ideals which are generated by monomials of degree $d$ in the polynomial ring in $n$ variables and which satisfy certain numerical side conditions regarding their exponents. Typical examples of such ideals are the ideals of Veronese…

We study a class of combinatorially-defined polynomial ideals which are generated by minors of a generic symmetric matrix. Included within this class are the symmetric determinantal ideals, the symmetric ladder determinantal ideals, and the…

The v-number of a graded ideal is an invariant recently introduced in the context of coding theory, particularly in the study of Reed--Muller-type codes. In this work, we study the localized v-numbers of a binomial edge ideal $J_G$…

This article proposes a bivariate polynomial problem for finite-order real matrices that endows a \textit{`sufficient condition'} for a map from the standard vector spaces of finite-order real matrices to the same dimensional bivariate…

We give a necessary and sufficient condition on a homogeneous polynomial ideal for its Taylor complex to be exact. Then we give a combinatorial construction of a minimal resolution for ideals satisfying the above condition (in particular…

Let J be a strongly stable monomial ideal in P=k[X0,...,Xn] and let BSt(J) be the family of all the homogeneous ideals in P such that the set N(J) of all the monomials that do not belong to J is a k-vector basis of the quotient P/I. We show…

We present an encoding of a polynomial system into vanishing and non-vanishing constraints on almost-principal minors of a symmetric, principally regular matrix, such that the solvability of the system over some field is equivalent to the…

Let $R=k[x,y,z]$ be a standard graded $3$-variable polynomial ring, where $k$ denotes any field. We study grade $3$ homogeneous ideals $I \subseteq R$ defining compressed rings with socle $k(-s)^{\ell} \oplus k(-2s+1)$, where $s \geq3$ and…

Let $\Lambda$ be a commutative Noetherian ring, and let $I$ be a proper ideal of $\Lambda$, $R=\Lambda /I$. Consider the polynomial rings $T=\Lambda [x_1,...x_n]$ and $A=R[x_1,...,x_n]$. Suppose that linear equations are solvable in…

We construct type A partially-symmetric Macdonald polynomials $P_{(\lambda \mid \gamma)}$, where $\lambda \in \mathbb{Z}_{\geq 0}^{n-k}$ is a partition and $\gamma \in \mathbb{Z}_{\geq 0}^k$ is a composition. These are polynomials which are…