Depth and Regularity of Monomial Ideals via Polarization and Combinatorial Optimization

  • José Martínez-Bernal
  • Susan Morey
  • Rafael H. VillarrealEmail author
  • Carlos E. Vivares


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.


Depth Regularity Max-flow min-cut Clutter Edge ideal Monomial ideal Polarization 

Mathematics Subject Classification (2010)

Primary 13F20 Secondary 05C22 05E40 13H10 



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

Funding Information

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


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© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  1. 1.Departamento de MatemáticasCentro de Investigación y de Estudios Avanzados del IPNMexico CityMexico
  2. 2.Department of MathematicsTexas State UniversitySan MarcosUSA

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