On the relative size of toric bases

TL;DR

This paper investigates the relationships between different bases of toric ideals, showing that no polynomial bounds exist relating their sizes or degrees when comparing two such bases.

Contribution

It proves that for any two bases of a toric ideal, there is no polynomial bound on the size or degree of one basis in terms of the other.

Findings

No polynomial bounds exist between sizes of different bases.

No polynomial bounds exist between degrees of different bases.

The result applies to Graver, Groebner, Markov bases, and circuits.

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.