Related papers: Algebraic properties of universal squarefree lexse…

Given a square-free monomial ideal $I$ in a polynomial ring $R$ over a field $\mathbb{K}$, one can associate it with its LCM-lattice and its hypergraph. In this short note, we establish the connection between the LCM-lattice and the…

We explore connections between the generalized multiplicities of square-free monomial ideals and the combinatorial structure of the underlying hypergraphs using methods of commutative algebra and polyhedral geometry. For instance, we show…

This extends a theorem of Davenport and Erd\"os on sequences of rational integers to sequences of integral ideals in arbitrary number fields $K$. More precisely, we introduce a logarithmic density for sets of integral ideals in $K$ and…

We analyze properties of various square-free factorizations in greatest common divisor domains and domains satisfying the ascending chain condition for principal ideals.

Let $I\subset S=K[x_1,...,x_n]$ be a lexsegment edge ideal or the Alexander dual of such an ideal. In both cases it turns out that the arithmetical rank of $I$ is equal to the projective dimension of $S/I.$

To a simplicial complex, we associate a square-free monomial ideal in the polynomial ring generated by its vertex set over a field. We study algebraic properties of this ideal via combinatorial properties of the simplicial complex. By…

In this paper I consider polynomial composites with the coefficients from $K\subset L$. We already know many properties, but we do not know the answer to the question of whether there is a relationship between composites and field…

We study representation of square-free polynomials in the polynomial ring F[t] over a finite field F by polynomials in F[t][x]. This is a function field version of the well-studied problem of representing squarefree integers by integer…

Let $S=K[x_1,\dots,x_n]$ be the polynomial ring over a field $K$ and $I\subset S$ be a squarefree monomial ideal generated in degree $n-2$. Motivated by the remarkable behavior of the powers of $I$ when $I$ admits a linear resolution, as…

We study, in a global uniform manner, the quotient of the ring of polynomials in l sets of n variables, by the ideal generated by diagonal quasi-invariant polynomials for general permutation groups W=G(r,n). We show that, for each such…

In this work, free multivariate skew polynomial rings are considered, together with their quotients over ideals of skew polynomials that vanish at every point (which includes minimal multivariate skew polynomial rings). We provide a full…

We consider powers of lexsegment ideals with a linear resolution (equivalently, with linear quotients) which are not completely lexsegment ideals. We give a complete description of their minimal graded free resolution.

Let $S={\sf k}[X_1,\dots, X_n]$ be a polynomial ring, where ${\sf k}$ is a field. This article deals with the defining ideal of the Rees algebra of squarefree monomial ideal generated in degree $n-2$. As a consequence, we prove that Betti…

Let $S=K[x_1,\ldots,x_n]$ be the polynomial ring over a field and $A$ a standard graded $S$-algebra. In terms of the Gr\"obner basis of the defining ideal $J$ of $A$ we give a condition, called the x-condition, which implies that all graded…

We develop a method for determining the density of squarefree values taken by certain multivariate integer polynomials that are invariants for the action of an algebraic group on a vector space. The method is shown to apply to the…

We introduce a family of squarefree monomial ideals associated to finite simple graphs, whose monomial generators correspond to closed neighborhood of vertices of the underlying graph. Any such ideal is called the closed neighborhood ideal…

We survey research relating algebraic properties of powers of squarefree monomial ideals to combinatorial structures. In particular, we describe how to detect important properties of (hyper)graphs by solving ideal membership problems and…

The associated primes of an arbitrary lexsegment ideal $I\subset S=K[x_1,...,x_n]$ are determined. As application it is shown that $S/I$ is a pretty clean module, therefore, $S/I$ is sequentially Cohen-Macaulay and satisfies Stanley's…

Let $S=K[x_1,\ldots,x_n]$ be the polynomial ring in $n$ variables over a field $K$ and $I\subset S$ a squarefree monomial ideal. In the present paper we are interested in the monomials $u \in S$ belonging to the socle $\Soc(S/I^{k})$ of…

In this paper we introduce the concepts of arbitrary $t$-spread lexsegments and of arbitrary $t$-spread lexsegment ideals with $t$ a positive integer. These concepts are a natural generalization of arbitrary lexsegments and arbitrary…