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


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.


Monomial ideal Freiman ideal Freiman graph Freiman matroid fiber cone 

Mathematics Subject Classification

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



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.


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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

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