Related papers: Generalized mixed product ideals whose powers have…

For positive integers m >= n >= p, we compute the GL_m x GL_n-equivariant description of the local cohomology modules of the polynomial ring S of functions on the space of m x n matrices, with support in the ideal of p x p minors. Our…

In this paper we give a necessary and sufficient combinatorial condition for a monomial ideal to have a linear resolution over fields of characteristic 2. We also give a new proof of Fr\"oberg's theorem over fields of characteristic 2.

We provide fundamental results on positive solutions to parametrized systems of generalized polynomial $\textit{inequalities}$ (with real exponents and positive parameters), including generalized polynomial $\textit{equations}$. In doing…

Let $S={\Bbb K}[x_1,\dots,x_n]$ denote a polynomial ring over a field $\Bbb K$. Given a monomial ideal $I$ and a finitely generated multigraded $M$ over $S$, we follow Herzog's method to construct a multigraded free $S$-resolution of $M/IM$…

We introduce the class of principal symmetric ideals, which are ideals generated by the orbit of a single polynomial under the action of the symmetric group. Fixing the degree of the generating polynomial, this class of ideals is…

We give criteria for graded ideals to have the property that all their powers are componentwise linear. Typical examples to which our criteria can be applied include the vertex cover ideals of certain finite graphs.

Let $R=K[x_1,...,x_n]$ be the polynomial ring in $n$ variables over a field $K$ and let $J$ be a matroidal ideal of degree $d$ in $R$. In this paper, we study the class of sequentially Cohen-Macaulay matroidal ideals. In particular, all…

Let $R=\mathbf{C}[\xi_1,\xi_2,\ldots]$ be the infinite variable polynomial ring, equipped with the natural action of the infinite symmetric group $\mathfrak{S}$. We classify the $\mathfrak{S}$-primes of $R$, determine the containments among…

Let $G$ be a finite simple graph and $J(G)$ denote its cover ideal in a polynomial ring over a field $\mathbb{K}$. In this paper, we show that all symbolic powers of cover ideals of certain vertex decomposable graphs have linear quotients.…

In this paper, we introduce techniques for producing normal square-free monomial ideals from old such ideals. These techniques are then used to investigate the normality of cover ideals under some graph operations. Square-free monomial…

For an ideal $I$ in a polynomial ring over a field, a monomial support of $I$ is the set of monomials that appear as terms in a set of minimal generators of $I$. Craig Huneke asked whether the size of a monomial support is a bound for the…

We consider ideals of polynomials vanishing on the W-orbits of the intersections of mirrors of a finite reflection group W. We determine all such ideals which are invariant under the action of the corresponding rational Cherednik algebra…

Let S=K[x_1,...,x_n] be a polynomial ring. Denote by $p_a$ the power sum symmetric polynomial x_1^a+...+x_n^a. We consider the following two questions: Describe the subsets $A \subset \mathbb{N}$ such that the set of polynomials $p_a$ with…

We develop an algebraic theory of supports for $R$-linear codes of fixed length, where $R$ is a finite commutative unitary ring. A support naturally induces a notion of generalized weights and allows one to associate a monomial ideal to a…

We survey some of the major results about normal Hilbert polynomials of ideals. We discuss a formula due to Lipman for complete ideals in regular local rings of dimension two, theorems of Huneke, Itoh, Huckaba, Marley and Rees in…

We will study monomial ideals $I$ in the exterior algebra as well as in the polynomial ring whose generic initial ideal is constant for all term orders up to permutations of variables. First, in the exterior algebra, we determine all graphs…

The content of a polynomial $f(t)$ is the ideal generated by its coefficients. Our aim here is to consider a beautiful formula of Dedekind-Mertens on the content of the product of two polynomials, to explain some of its features from the…

If I is an ideal in a Gorenstein ring S and S/I is Cohen-Macaulay, then the same is true for any linked ideal I'. However, such statements hold for residual intersections of higher codimension only under very restrictive hypotheses, not…

We describe the ideals, especially the prime ideals, of semirings of polynomials over layered domains, and in particular over supertropical domains. Since there are so many of them, special attention is paid to the ideals arising from…

We show that any lexsegment ideal with linear resolution has linear quotients with respect to a suitable ordering of its minimal monomial generators. For completely lexsegment ideals with linear resolution we show that the decomposition…