Acta Mathematica Vietnamica

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Depth and Regularity of Monomial Ideals via Polarization and Combinatorial Optimization

José Martínez-Bernal , Susan Morey , $icon-email$ Rafael H. Villarreal , Carlos E. Vivares

Abstract

In this paper, we use polarization to study the behavior of the depth and regularity of a monomial ideal $I$ , locally at a variable $x_{i}$ , when we lower the degree of all the highest powers of the variable $x_{i}$ occurring in the minimal generating set of $I$ , and examine the depth and regularity of powers of edge ideals of clutters using combinatorial optimization techniques. If $I$ is the edge ideal of an unmixed clutter with the max-flow min-cut property, we show that the powers of $I$ have non-increasing depth and non-decreasing regularity. In particular, edge ideals of unmixed bipartite graphs have non-decreasing regularity. We are able to show that the symbolic powers of the ideal of covers of the clique clutter of a strongly perfect graph have non-increasing depth. A similar result holds for the ideal of covers of a uniform ideal clutter.

Acta Mathematica Vietnamica

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Depth and Regularity of Monomial Ideals via Polarization and Combinatorial Optimization

Abstract