Related papers: Groebner bases via linkage

The interpolation step of Guruswami and Sudan's list decoding of Reed-Solomon codes poses the problem of finding the minimal polynomial of an ideal with respect to a certain monomial order. An efficient algorithm that solves the problem is…

We give a necessary condition for the existence of a path of n irreducible morphisms between indecomposable modules whose composition lies in the (n + 1)-power of the radical. In order to do that, we consider the general criterion given by…

In this paper, we consider a monomial ideal J in P := A[x1,...,xn], over a commutative ring A, and we face the problem of the characterization for the family Mf(J) of all homogeneous ideals I in P such that the A-module P/I is free with…

The aim of this note is to discuss the following quite queer Problem: \noindent GIVEN \noindent i) the free non-commutative polynomial ring, ${\Cal P} := {\Bbb F}\langle X_1,\ldots,X_n\rangle$ {\em (public)}, \noindent ii) a bilateral ideal…

Describing families of ideals that are minimally generated by at least one, or by all, of their reduced Gr\"obner bases is a central topic in commutative algebra. In this paper, we address this problem in the context of toric ideals of…

We consider ordered pairs $(X,\mathcal{B})$ where $X$ is a finite set of size $v$ and $\mathcal{B}$ is some collection of $k$-element subsets of $X$ such that every $t$-element subset of $X$ is contained in exactly $\lambda$ "blocks" $B\in…

Let $K$ be a field and $R=K[x_1,\ldots, x_n]$ be the polynomial ring in $n$ variables over a field $K$. Let $\Delta$ be a simplicial complex on $n$ vertices and $I=I_{\Delta}$ be its Stanley-Reisner ideal. In this paper, we show that if $I$…

Monomial ideals which are generic with respect to either their generators or irreducible components have minimal free resolutions derived from simplicial complexes. For a generic monomial ideal, the associated primes satisfy a saturated…

Let $(R,\mathfrak{m},\Bbbk)$ be a regular local ring of dimension 3. Let $I$ be a Gorenstein ideal of $R$ of grade 3. It follows from a result of Buchsbaum and Eisenbud that there is a skew-symmetric matrix of odd size such that $I$ is…

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…

Solving multihomogeneous systems, as a wide range of structured algebraic systems occurring frequently in practical problems, is of first importance. Experimentally, solving these systems with Gr\"obner bases algorithms seems to be easier…

In this paper, we draw connections between ideal lattices and multivariate polynomial rings over integers using Gr\"obner bases. Ideal lattices are ideals in the residue class ring, $\mathbb{Z}[x]/\langle f \rangle$ (here $f$ is a monic…

We present a $p$-adic algorithm to recover the lexicographic Gr\"obner basis $\mathcal G$ of an ideal in $\mathbb Q[x,y]$ with a generating set in $\mathbb Z[x,y]$, with a complexity that is less than cubic in terms of the dimension of…

We study the complexity of Gr\"obner bases computation, in particular in the generic situation where the variables are in simultaneous Noether position with respect to the system. We give a bound on the number of polynomials of degree $d$…

Brambles were introduced as the dual notion to treewidth, one of the most central concepts of the graph minor theory of Robertson and Seymour. Recently, Grohe and Marx showed that there are graphs G, in which every bramble of order larger…

Conca and Varbaro (Invent. Math. 221 (2020), no. 3) showed the equality of depth of a graded ideal and its initial ideal in a polynomial ring when the initial ideal is square-free. In this paper, we give some beautiful applications of this…

A minor is principal means it is defined by the same row and column indices. Let $X$ be a square generic matrix, $K[X]$ the polynomial ring in entries of $X$, over an algebraically closed field, $K$. For fixed $t\leq n$, let $\mathfrak P_t$…

Radical binomial ideals associated with finite lattices are studied. Gr\"obner basis theory turns out to be an efficient tool in this investigation.

We consider the problem of computing a grevlex Gr\"obner basis for the set $F_r(M)$ of minors of size $r$ of an $n\times n$ matrix $M$ of generic linear forms over a field of characteristic zero or large enough. Such sets are not regular…

We focus on Gr\"obner bases for modules of univariate polynomial vectors over a ring. We identify a useful property, the "predictable leading monomial (PLM) property" that is shared by minimal Gr\"{o}bner bases of modules in F[x]^q, no…