Advertisement

Mathematische Zeitschrift

, Volume 291, Issue 3–4, pp 999–1014 | Cite as

The relevance of Freiman’s theorem for combinatorial commutative algebra

  • Jürgen Herzog
  • Takayuki Hibi
  • Guangjun ZhuEmail author
Article
  • 55 Downloads

Abstract

Freiman’s theorem gives a lower bound for the cardinality of the doubling of a finite set in \({\mathbb R}^n\). In this paper we give an interpretation of his theorem for monomial ideals and their fiber cones. We call a quasi-equigenerated monomial ideal a Freiman ideal, if the set of its exponent vectors achieves Freiman’s lower bound for its doubling. Algebraic characterizations of Freiman ideals are given, and finite simple graphs are classified whose edge ideals or matroidal ideals of its cycle matroids are Freiman ideals.

Keywords

Monomial ideal Freiman ideal Freiman graph Freiman matroid fiber cone 

Mathematics Subject Classification

Primary 13C99 Secondary 13A15 13E15 13H05 13H10 

Notes

Acknowledgements

This paper is supported by the National Natural Science Foundation of China (11271275) and by the Foundation of the Priority Academic Program Development of Jiangsu Higher Education Institutions. We would like to thank the referee for a careful reading and pertinent comments.

References

  1. 1.
    Abhyankar, S.S.: Local rings of high embedding dimension. Am. J. Math. 89, 1073–1077 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Blasiak, J.: The toric ideal of a graphic matroid is generated by quadrics. Combinatorica 28, 283–297 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bruns, W., Herzog, J.: Cohen–Macaulay Rings. Cambridge University Press, Cambridge (1998)CrossRefzbMATHGoogle Scholar
  4. 4.
    Böröczky, K.J., Santos, F., Serra, O.: On sumsets and convex hull. Discr. Comput. Geom. 52, 705–729 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Eisenbud, D., Goto, S.: Linear free resolutions and minimal multiplicity. J. Algebra 88, 89–133 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Eagon, J.A., Northcott, D.G.: Ideals defined by matrices and a certain complex associated with them. Proc. R. Soc. Lond. Ser. A 269, 188–204 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Freiman, G.A.: Foundations of a structural theory of set addition, Translations of mathematical monographs 37. American Mathematical Society, Providence, Phode Island (1973)Google Scholar
  8. 8.
    Ge, M., Lin, J., Wang, Y.: Hilbert series and Hilbert depth of squarefree Veronese ideals. J. Algebra 344, 260–267 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Goto, S., Watanabe, K.: On graded rings, I. J. Math. Soc. Jpn. 30, 179–213 (1978)CrossRefzbMATHGoogle Scholar
  10. 10.
    Herzog, J., Hibi, T.: Monomial Ideals. Graduate Texts in Mathematics, vol. 260. Springer, London (2010)zbMATHGoogle Scholar
  11. 11.
    Herzog, J., Mohammadi Saem, M., Zamani, N.: On the number of generators of powers of an ideal. arXiv:1707.07302v1
  12. 12.
    Herzog, J., Zhu, G.J.: Freiman ideals. Comm. Algebra. arXiv:1709.02827v1 (to appear)
  13. 13.
    Hibi, T.: Algebraic Combinatorics on Convex Polytopes. Carslaw Publications, Glebe, N. S. W, Australia (1992)zbMATHGoogle Scholar
  14. 14.
    Hoa, L.T., Tam, N.D.: On some invariants of a mixed product of ideals. Arch. Math. 94, 327–337 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Hochster, M.: Rings of invariants of tori, Cohen–Macaulay rings generated by monomials, and polytopes. Ann. Math. 96, 228–235 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Ohsugi, H., Herzog, J., Hibi, T.: Combinatorial pure subrings. Osaka J. Math. 37, 745–757 (2000)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Ohsugi, H., Hibi, T.: Toric Ideals generated by Quadratic binomials. J. Algebra 218, 509–527 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Sally, J.D.: On the associated graded rings of a local Cohen–Macaulay ring. J. Math. Kyoto Univ. 17, 19–21 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Shah, K.: On the Cohen–Macaulayness of the fiber cone of an ideal. J. Algebra 143, 156–172 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Stanchescu, Y.V.: On the simplest inverse problem for sums of sets in several dimensions. Combinatorica 18, 139–149 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    White, N.L.: The basis monomial ring of a matroid. Adv. Math. 24, 292–297 (1977)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Fachbereich MathematikUniversität Duisburg-EssenEssenGermany
  2. 2.Department of Pure and Applied Mathematics, Graduate School of Information Science and TechnologyOsaka UniversityToyonakaJapan
  3. 3.School of Mathematical SciencesSoochow UniversitySuzhouPeople’s Republic of China

Personalised recommendations