On the relative size of toric bases

Abstract

We consider the Graver basis, the universal Groebner 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 $A\not \subset B$, 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.