On the relative size of toric bases

Abstract

We consider the Graver basis, the universal Gröbner basis, a Markov basis and the set of the circuits of a toric ideal. Let A, B be any two of these bases such that A ⊂ B, we prove that there is no polynomial on the size or on the maximal degree of the elements of B which bounds the size or the maximal degree of the elements of A correspondingly.