Membership Criteria and Containments of Powers of Monomial Ideals


We present a close relationship between matching number, covering numbers and their fractional versions in combinatorial optimization and ordinary powers, integral closures of powers, and symbolic powers of monomial ideals. This relationship leads to several new results and problems on the containments between these powers.

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    Our examples were obtained independently, and that was communicated to Huneke on August 8, 2017.

  2. 2.

    The authors thank Adam Van Tuyl for pointing them to this inequality.


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This paper started during a research stay of the authors at Vietnam Institute for Advanced Study in Mathematics. The authors would like to thank the institute for its support and hospitality.


The first author is partially supported by Simons Foundation (grant # 279786) and Louisiana Board of Regents (grant # LEQSF(2017-19)-ENH-TR-25). The second author is supported by Vietnam National Foundation for Science and Technology Development (grant # 101.04-2017.19).

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Correspondence to Ngo Viet Trung.

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Dedicated to Le Tuan Hoa on the occasion of his 60th birthday

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Hà, H.T., Trung, N.V. Membership Criteria and Containments of Powers of Monomial Ideals. Acta Math Vietnam 44, 117–139 (2019).

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  • Monomial ideal
  • Ordinary power
  • Symbolic power
  • Integral closure of a power
  • Hypergraph
  • Matching
  • Covering
  • Gap estimate
  • Edge ideal
  • Containments between powers of ideals
  • Generating degree

Mathematics Subject Classification (2010)

  • 13C05
  • 05C65
  • 90C27