Related papers: Alexander duality for monomial ideals associated w…

We exhibit several posets arising from commutative algebra, order theory, tropical convexity as potential face posets of tropical polyhedra, and we clarify their inclusion relations. We focus on monomial tropical polyhedra, and deduce how…

Proofs of two results about a monomial ideal -- describing membership in auxiliary ideals associated to the monomial ideal -- are given which do not invoke resolution of singularities. The AM--GM inequality is used as a substitute for…

We study the Macaulay coefficients induced by the ideal and quotient segments of a degree-$\delta$ monomial in $n$ variables. We give explicit formulas for these coefficients and establish a duality between the two theories. Our main result…

A classical result due to Morita and Azumaya establishes that given two arbitrary rings, any duality between their finitely generated modules is representable by a faithfully balanced bimodule which is a finitely generated injective…

The aim of this paper is to derive (by using two operators, representable by a Jacobi matrix) a family of q-orthogonal polynomials, which turn to be dual to alternative q-Charlier polynomials. A discrete orthogonality relation and a…

Alexander polynomials of sextics with only simple singularities or sextics of torus type with arbitrary singularities are computed. We show that for ieeducible sextics,there are four possibilities: $(t^2-t+1)^j, j=0,1,2,3$.

Let $M$ be a T-motive. We introduce the notion of duality for $M$. Main results of the paper (we consider uniformizable $M$ over $F_q[T]$ of rank $r$, dimension $n$, whose nilpotent operator $N$ is 0): 1. Algebraic duality implies analytic…

This paper uses dualities between facet ideal theory and Stanley-Reisner theory to show that the facet ideal of a simplicial tree is sequentially Cohen-Macaulay. The proof involves showing that the Alexander dual (or the cover dual, as we…

This paper extends Alexander duality to the setting of parametrized homology. Let X with be a compact subset of R^n x R (n \geq 2) satisfying certain conditions, let Y be its complement, and let p be the projection onto the second factor.…

The aim of this article is to show that the transpose-dual pairs in the sense of Ebeling-Ploog of singularities $(Z_{1,0},\, Z_{1,0})$, $(U_{1,0},\, U_{1,0})$, $(Q_{17},\, Z_{2,0})$, $(W_{1,0},\, W_{1,0})$ that are concluded to be…

Irreducible decomposition of monomial ideals has an increasing number of applications from biology to pure math. This paper presents the Slice Algorithm for computing irreducible decompositions, Alexander duals and socles of monomial…

We partially prove and partially disprove Oka's conjecture on the fundamental group/Alexander polynomial of an irreducible plane sextic. Among other results, we enumerate all irreducible sextics with simple singularities admitting dihedral…

We give a necessary and sufficient condition for a simplicial complex to be approximately Cohen-Macaulay. Namely it is approximately Cohen-Macaulay if and only if the ideal associated to its Alexander dual is componentwise linear and…

Let $L$ be a separable quadratic extension of either $\mathbb{Q}$ or $\mathbb{F}_q(t)$. We propose efficient algorithms for finding isomorphisms between quaternion algebras over $L$. Our techniques are based on computing maximal one-sided…

A recent pair of papers of Armstrong, Loehr, and Warrington and Armstrong, Williams, and the author initiated the systematic study of {\em rational Catalan combinatorics} which is a generalization of Fuss-Catalan combinatorics (which is in…

An ideal of polynomials is symmetric if it is closed under permutations of variables. We relate general symmetric ideals to the so called Specht ideals generated by all Specht polynomials of a given shape. We show a connection between the…

In this paper we give new upper bounds on the regularity of edge ideals whose resolutions are k-steps linear; surprisingly, the bounds are logarithmic in the number of variables. We also give various bounds for the projective dimension of…

We investigate, using the notion of linear quotients, significative classes of connected graphs whose monomial edge ideals, not necessarily squarefree, have linear resolution, in order to compute standard algebraic invariants of the…

We prove an Alexander-type duality for valuations for certain subcomplexes in the boundary of polyhedra. These strengthen and simplify results of Stanley (1974) and Miller-Reiner (2005). We give a generalization of Brion's theorem for this…

We give a necessary and sufficient condition on a homogeneous polynomial ideal for its Taylor complex to be exact. Then we give a combinatorial construction of a minimal resolution for ideals satisfying the above condition (in particular…