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Powers of Square-Free Monomial Ideals and Combinatorics

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Abstract

We survey research relating algebraic properties of powers of squarefree monomial ideals to combinatorial structures. In particular, we describe how to detect important properties of (hyper)graphs by solving ideal membership problems and computing associated primes. This work leads to algebraic characterizations of perfect graphs independent of the Strong Perfect Graph Theorem. In addition, we discuss the equivalence between the Conforti-Cornuéjols conjecture from linear programming and the question of when symbolic and ordinary powers of squarefree monomial ideals coincide.

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Acknowledgements

This work was partially supported by grants from the Simons Foundation (#199124 to Francisco and #202115 to Mermin). Hà is partially supported by NSA grant H98230-11-1-0165.

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Authors and Affiliations

  1. Department of Mathematics, Oklahoma State University, 401 Mathematical Sciences, Stillwater, OK, 74078, USA

    Christopher A. Francisco & Jeffrey Mermin

  2. Department of Mathematics, Tulane University, 6823 St. Charles Ave., New Orleans, LA, 70118, USA

    Huy Tài Hà

Authors
  1. Christopher A. Francisco
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  2. Huy Tài Hà
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  3. Jeffrey Mermin
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Correspondence to Jeffrey Mermin .

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Editors and Affiliations

  1. Cornell University, 547 Malott Hall, Ithaca, 14853, New York, USA

    Irena Peeva

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Francisco, C.A., Hà, H.T., Mermin, J. (2013). Powers of Square-Free Monomial Ideals and Combinatorics. In: Peeva, I. (eds) Commutative Algebra. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-5292-8_11

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