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Depth and Regularity of Monomial Ideals via Polarization and Combinatorial Optimization

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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 xi, when we lower the degree of all the highest powers of the variable xi 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.

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Acknowledgements

We thank the referee for a careful reading of the paper and for the improvements suggested.

Funding

The first and third authors were partially supported by SNI. The fourth author was supported by a scholarship from CONACYT.

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Authors and Affiliations

  1. Departamento de Matemáticas, Centro de Investigación y de Estudios Avanzados del IPN, Apartado Postal 14–740, 07000, Mexico City, D.F., Mexico

    José Martínez-Bernal, Rafael H. Villarreal & Carlos E. Vivares

  2. Department of Mathematics, Texas State University, San Marcos, TX, 78666, USA

    Susan Morey

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  1. José Martínez-Bernal
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  2. Susan Morey
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  3. Rafael H. Villarreal
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Correspondence to Rafael H. Villarreal.

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Martínez-Bernal, J., Morey, S., Villarreal, R.H. et al. Depth and Regularity of Monomial Ideals via Polarization and Combinatorial Optimization. Acta Math Vietnam 44, 243–268 (2019). https://doi.org/10.1007/s40306-018-00308-z

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