The relevance of Freiman’s theorem for combinatorial commutative algebra

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.