Depth and Regularity of Monomial Ideals via Polarization and Combinatorial Optimization

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 power...

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Bibliographic Details
Published inActa mathematica vietnamica Vol. 44; no. 1; pp. 243 - 268
Main Authors Martínez-Bernal, José, Morey, Susan, Villarreal, Rafael H., Vivares, Carlos E.
Format Journal Article
LanguageEnglish
Published Singapore Springer Singapore 15.03.2019
Springer Nature B.V
Subjects
Clutter
Combinatorial analysis
Mathematics
Mathematics and Statistics
Optimization
Optimization techniques
Polarization
Regularity
Monomial ideal
Polarization
05E40
Max-flow min-cut
Edge ideal
Primary 13F20
Secondary 05C22
Regularity
Clutter
Depth
13H10
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Summary: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.
ISSN:0251-4184
2315-4144
DOI:10.1007/s40306-018-00308-z