Ideals of fiber type and polymatroids

Abstract

In the first half of this paper, we complement the theory on discrete polymatroids. More precisely, (i) we prove that a discrete polymatroid satisfying the strong exchange property is, up to an affinity, of Veronese type; (ii) we classify all uniform matroids which are level; (iii) we introduce the concept of ideals of fiber type and show that all polymatroidal ideals are of fiber type. On the other hand, in the latter half of this paper, we generalize the result proved by Stefan Blum that the defining ideal of the Rees ring of a base sortable matroid possesses a quadratic Gröbner basis. For this purpose we introduce the concept of ``$l$-exchange property'' and show that a Gröbner basis of the defining ideal of the Rees ring of an ideal $I$ can be determined and that $I$ is of fiber type if $I$ satisfies the $l$-exchange property. Ideals satisfying the $l$-exchange property include strongly stable ideals, polymatroid ideals of base sortable discrete polymatroids, ideals of Segre-Veronese type and certain ideals related to classical root systems.