Mathematische Zeitschrift

, Volume 291, Issue 3–4, pp 999–1014

# The relevance of Freiman’s theorem for combinatorial commutative algebra

• Jürgen Herzog
• Takayuki Hibi
• Guangjun Zhu
Article

## 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)
2. 2.
Blasiak, J.: The toric ideal of a graphic matroid is generated by quadrics. Combinatorica 28, 283–297 (2008)
3. 3.
Bruns, W., Herzog, J.: Cohen–Macaulay Rings. Cambridge University Press, Cambridge (1998)
4. 4.
Böröczky, K.J., Santos, F., Serra, O.: On sumsets and convex hull. Discr. Comput. Geom. 52, 705–729 (2014)
5. 5.
Eisenbud, D., Goto, S.: Linear free resolutions and minimal multiplicity. J. Algebra 88, 89–133 (1984)
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)
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)
9. 9.
Goto, S., Watanabe, K.: On graded rings, I. J. Math. Soc. Jpn. 30, 179–213 (1978)
10. 10.
Herzog, J., Hibi, T.: Monomial Ideals. Graduate Texts in Mathematics, vol. 260. Springer, London (2010)
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)
14. 14.
Hoa, L.T., Tam, N.D.: On some invariants of a mixed product of ideals. Arch. Math. 94, 327–337 (2010)
15. 15.
Hochster, M.: Rings of invariants of tori, Cohen–Macaulay rings generated by monomials, and polytopes. Ann. Math. 96, 228–235 (1972)
16. 16.
Ohsugi, H., Herzog, J., Hibi, T.: Combinatorial pure subrings. Osaka J. Math. 37, 745–757 (2000)
17. 17.
Ohsugi, H., Hibi, T.: Toric Ideals generated by Quadratic binomials. J. Algebra 218, 509–527 (1999)
18. 18.
Sally, J.D.: On the associated graded rings of a local Cohen–Macaulay ring. J. Math. Kyoto Univ. 17, 19–21 (1977)
19. 19.
Shah, K.: On the Cohen–Macaulayness of the fiber cone of an ideal. J. Algebra 143, 156–172 (1991)
20. 20.
Stanchescu, Y.V.: On the simplest inverse problem for sums of sets in several dimensions. Combinatorica 18, 139–149 (1998)
21. 21.
White, N.L.: The basis monomial ring of a matroid. Adv. Math. 24, 292–297 (1977)