In this paper, we consider a monomial ideal J / P := A[x1, . . . , xn], over a commutative ring A, and we face the problem of the characterization for the familyMf (J) of all homogeneous ideals I / P such that the A-module P/I is free with basis given by the set of terms in the Gröbner escalier N(J) of J. This family is in general wider than that of the ideals having J as initial ideal w.r.t. a...

A recent paper by Amiraslani, Corless, Gonzalez-Vega and Shakoori studies polynomial algebra by values, without first converting to another basis such as the monomial basis. In this talk I expand on some details from that paper, namely the method we used to divide (multivariate and univariate) polynomials given only by values. This is a surprisingly valuable operation, and with it one can solve...

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 monomial set. Such a division provides for each monomial the self-consistent separation of the whole set of variables into two disjoint subsets. They are cal...

We call a graded connected algebra R effectively coherent, if for every linear equation over R with homogeneous coefficients of degrees at most d, the degrees of generators of its module of solutions are bounded by some function D(d). For commutative polynomial rings, this property has been established by Hermann in 1926. We establish the same property for several classes of noncommutative alge...

We study a graded algebra D=D(L,G) defined by a finite lattice L and a subset G in L, a so-called building set. This algebra is a generalization of the cohomology algebras of hyperplane arrangement compactifications found in work of De Concini and Procesi [D2]. Our main result is a representation of D, for an arbitrary atomic lattice L, as the Chow ring of a smooth toric variety that we constru...

G. E. Murphy showed in 1983 that the centre of every symmetric group algebra has an integral basis consisting of a specific set of monomial symmetric polynomials in the Jucys–Murphy elements. While we have shown in earlier work that the centre of the group algebra of S3 has exactly three additional such bases, we show in this paper that the centre of the group algebra of S4 has infinitely many ...

This is the first in a series of papers on standard monomial theory and invariant arc spaces. For any algebraically closed field $K$, we construct basis for space determinantal variety over $K$. As an application, prove analogue second fundamental theorems general linear group.

In this article, we consider the matrix equation $AXB=C$, where A, B, C are given matrices and give new necessary and sufficient conditions for the existence of the diagonal solutions and monomial solutions to this equation. We also present a general form of such solutions. Moreover, we consider the least squares problem $min_X |C-AXB |_F$ where $X$ is a diagonal or monomial matrix. The explici...

Let L be the generalized mixed product ideal induced by a monomial ideal I. In this paper we compute powers of the genearlized mixed product ideals and show that Lk  have a linear resolution if and only if Ik have a linear resolution for all k. We also introduce the generalized mixed polymatroidal ideals and prove that powers and monomial localizations of a generalized mixed polymatroidal ideal...

Let I be a monomial ideal in a polynomial ring over a field k. If the Rees cone of I is quasi-ideal, we express the normalization of the Rees algebra of I in terms of an Ehrhart ring. We introduce the basis Rees cone of a matroid and study their facets. Then an application to Rees algebras is presented.