Membership Criteria and Containments of Powers of Monomial Ideals

Abstract

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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Notes

  1. 1.

    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.

References

  1. 1.

    Asgharzadeh, M.: Huneke’s degree-computing problem. arXiv:1609.04323.v2 (2017)

  2. 2.

    Bauer, T., Di Rocco, S., Harbourne, B., Kapustka, M.L., Knutsen, A., Syzdek, W., Szemberg, T.: A primer on Seshadri constants. Contemp. Math. 496, 33–70 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Baum, S., Trotter, L.E.: Integer rounding for polymatroid and branching optimization problems. SIAM J. Algebraic Discrete Method 2, 416–425 (1981)

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Baum, S., Trotter, L.E.: Finite checkability for integer rounding properties in combinatorial programming problems. Math. Programming 22, 141–147 (1982)

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Bocci, C., Cooper, S., Guardo, E., Harbourne, B., Janssen, M., Nagel, U., Seceleanu, A., Van Tuyl, A., Vu, T.: The Waldschmidt constant for squarefree monomial ideals. J. Algebraic Combin. 44(4), 875–904 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Bocci, C., Harbourne, B.: Comparing powers and symbolic powers of ideals. J. Algebraic Geom. 19, 399–417 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Conforti, M., Cornuéjols, G.: Clutters that pack and the max flow min cut property: a conjecture, Management Science Research Report. http://handle.dtic.mil/100.2/ADA277340 (1993)

  8. 8.

    Cooper, S., Embree, R.J.D., Hà, H.T., Hoefel, A.H.: Symbolic powers of monomial ideals. Proc. Edinb. Math. Soc. (2) 60(1), 39–55 (2017)

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Dao, H., De Stefani, A., Grifo, E., Huneke, C., Núñez-Betancourt, L.: Symbolic Powers of Ideals. In: Singularities and Foliations, Geometry, Topology and Applications, 387–432, Springer Proc. Math. Stat. 222 (2018)

  10. 10.

    Dipasquale, M., Francisco, C.A., Mermin, J., Schweig, J.: Asymptotic resurgence via integral closures. arXiv:1808.01547 (2018)

  11. 11.

    Duchet, P.: Hypergraphs. In: Graham, R., Grötschel, M., Lovász, L. (eds.) Handbook of Combinatorics (1995)

  12. 12.

    Dupont, L.A., Villarreal, R.H.: Edge ideals of clique clutters of comparability graphs and the normality of monomial ideals. Math. Scand. 106, 88–98 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Ein, L., Lazarsfeld, R., Smith, K.E.: Uniform bounds and symbolic powers on smooth varieties. Invent. Math. 144, 241–252 (2001)

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Francisco, C.A., Hà, H.T., Mermin, J.: Powers of Square-Free Monomial Ideals and Combinatorics. In: Commutative Algebra, pp 373–392. Springer (2013)

  15. 15.

    Füredi, Z.: Maximum degrees and fractional matchings in uniform hypergraphs. Combinatorica 1, 154–162 (1981)

    MathSciNet  Google Scholar 

  16. 16.

    Füredi, Z., Kahn, J., Seymour, P.D.: On the fractional matching polytope of a hypergraph. Combinatorica 13, 167–180 (1993)

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    Gitler, I., Valencia, C.E., Villarreal, R.: A note on Rees algebras and the MFMC property. Contributions to Algebra and Geometry 48, 141–150 (2007)

    MathSciNet  MATH  Google Scholar 

  18. 18.

    Herzog, J., Hibi, T., Trung, N.V.: Symbolic powers of monomial ideals and vertex cover algebras. Adv. Math. 210(1), 304–322 (2007)

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    Herzog, J., Hibi, T., Trung, N.V., Zheng, X.: Standard graded vertex cover algebras, cycles and leaves. Trans. Am. Math. Soc. 360, 6231–6249 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    Henderson, J.R.: Permutation decompositions of (0,1)-matrices and decomposition transversals. Thesis, Caltech (1971)

  21. 21.

    Hochster, M., Huneke, C.: Comparison of symbolic and ordinary powers of ideals. Invent. Math. 147, 349–369 (2002)

    MathSciNet  Article  MATH  Google Scholar 

  22. 22.

    Huneke, C.: Uniform bounds in Noetherian rings. Invent. Math. 107(1), 203–223 (1992)

    MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    Huneke, C.: Open problems on powers of ideals, Notes for the Workshop on Integral Closure, Multiplier Ideals and Cores, AIM. https://www.aimath.org/WWN/integralclosure/Huneke.pdf (2006)

  24. 24.

    Lipman, J., Sathaye, A.: Jacobian ideals and a theorem of briançon-skoda. Michigan Math. J. 28, 199–222 (1981)

    MathSciNet  Article  MATH  Google Scholar 

  25. 25.

    Lipman, J., Teissier, B.: Pseudorational local rings and a theorem of briançon-skoda about integral closures of ideals. Michigan Math. J. 28, 97–116 (1981)

    MathSciNet  Article  MATH  Google Scholar 

  26. 26.

    Lovász, L.: On minimax theorems of combinatorics, PhD Thesis. Mathematikai Lapok 26, 209–264 (1975)

    Google Scholar 

  27. 27.

    Lovász, L.: On the ratio of optimal integral and fractional covers. Discrete Math. 13, 383–390 (1975)

    MathSciNet  Article  MATH  Google Scholar 

  28. 28.

    Ma, L., Schwede, K.: Perfectoid multiplier/test ideals in regular rings and bounds on symbolic powers. arXiv:1705.02300 (2017)

  29. 29.

    Medina, R., Noyer, C., Raynaud, O.: Twins vertices in hypergraphs. Electron. Notes Discrete Math. 27, 87–89 (2006)

    Article  MATH  Google Scholar 

  30. 30.

    Scheinerman, E.R., Ullman, D.H.: Fractional graph theory. A rational approach to the theory of graphs. With a foreword by Claude Berge. Reprint of the 1997 original. Dover Publications, Inc., Mineola, NY, 2011. xviii+ 211 pp.

  31. 31.

    Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency, vol. 1. Springer, Berlin (2003)

    Google Scholar 

  32. 32.

    Trung, N.V.: Integral closures of monomial ideals and Fulkersonian hypergraphs. Vietnam J. Math. 34, 489–494 (2006)

    MathSciNet  MATH  Google Scholar 

  33. 33.

    Trung, N.V.: Square-free monomial ideals and hypergraphs, Notes for the Workshop on Integral Closure, Multiplier Ideals and Cores, AIM. https://www.aimath.org/WWN/integralclosure/Trung.pdf (2006)

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Acknowledgments

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.

Funding

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). https://doi.org/10.1007/s40306-018-00325-y

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Keywords

  • 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