Related papers: Unmixed polymatroidal ideals

Let $R=k[x_1,...,x_n]$ be the polynomial ring in $n$ variables over a field $k$ and $I$ be a matroidal ideal of degree $d$. In this paper, we study the unmixedness properties and the arithmetical rank of $I$. Moreover, we show that…

Let $R=K[x_1,...,x_n]$ be the polynomial ring in $n$ variables over a field $K$ and $I$ be a monomial ideal generated in degree $d$. Bandari and Herzog conjectured that a monomial ideal $I$ is polymatroidal if and only if all its monomial…

This paper was motivated by a problem left by Herzog and Hibi, namely to classify all unmixed polymatroidal ideals. In the particular case of polymatroidal ideals corresponding to discrete polymatroids of Veronese type, i.e ideals of…

In this paper we study classes of monomial ideals for which all of its powers have a linear resolution. Let K[x_{1},x_{2}] be the polynomial ring in two variables over the field K, and let L be the generalized mixed product ideal induced by…

Let $R=K[x_1,\ldots, x_n]$ be the polynomial ring in $n$ variables over a field $K$ and let $M_{n,t}=(x^{e_1},\ldots, x^{e_n})$ be a monomial ideal of $R$, where $x^{e_i}=x_1^t\ldots x_{i-1}^tx_{i+1}^t\ldots x_n^t$. We study the unmixedness…

Let L be the generalized mixed product ideal induced by a monomial ideal I. In this paper, we study the polymatroidal property of generalized mixed product ideals. Furthermore, some algebraic invariants of L are computed.

We introduce the cave polynomial of a polymatroid and show that it yields a valuative function on polymatroids. The support of this polynomial after homogenization is again a polymatroid. The cave polynomial gives a $K$-theoretic…

Let $R = k[x_1,\ldots, x_n]$ be the polynomial ring in $n$ variables over a field $k$ and let $I$ be a monomial ideal of $R$. In this paper, we study almost Cohen-Macaulay simplicial complex. Moreover, we characterize the almost…

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…

We determine, in a polynomial ring over a field, the arithmetical rank of certain ideals generated by a set of monomials and one binomial.

For a given ideal I in K[x_1,...,x_n,y_1,...,y_m] in a polynomial ring with n+m variables, we want to find all elements that can be written as f-g for some f in K[x_1,...,x_n] and some g in K[y_1,...,y_m], i.e., all elements of I that…

We study the homological shifts of polymatroidal ideals. In our main theorem we prove that the first homological shift ideal of any polymatroidal ideal is again polymatroidal, supporting a conjecture of Bandari, Bayati and Herzog that…

We consider ideals in the ring $\mathbb{Z}_2[x_1,\ldots, x_n]$ that contain the polynomials $x_i^2 - x_i$ for $i = 1, \ldots, n$ and give various results related to the one-to-one correspondence between these ideals and the subsets of…

In this paper we consider monomial localizations of monomial ideals and conjecture that a monomial ideal is polymatroidal if and only if all its monomial localizations have a linear resolution. The conjecture is proved for squarefree…

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…

In this article we investigate when a homogeneous ideal in a graded ring is normal, that is, when all positive powers of the ideal are integrally closed. We are particularly interested in homogeneous ideals in an N-graded ring generated by…

Let $R = \mathbb{K}[x_1, \ldots, x_n]$ be a polynomial ring over a field $\mathbb{K}$, and let $I \subseteq R$ be a monomial ideal of height $h$. We provide a formula for the multiplicity of the powers of $I$ when all the primary ideals of…

Irreducible decompositions of monomial ideals in polynomial rings over a field are well-understood. In this paper, we investigate decompositions in the set of monomial ideals in the semigroup ring A[\mathbb{R}_{\geq 0}^d] where A is an…

The present paper investigates properties of quasi-stable ideals and of Borel-fixed ideals in a polynomial ring $k[x_0,\dots,x_n]$, in order to design two algorithms: the first one takes as input $n$ and an admissible Hilbert polynomial…

Regarding polynomial functions on a subset $S$ of a non-commutative ring $R$, that is, functions induced by polynomials in $R[x]$ (whose variable commutes with the coefficients), we show connections between, on one hand, sets $S$ such that…