Graver degrees are not polynomially bounded by true circuit degrees

TL;DR

This paper demonstrates that the degrees of elements in the Graver basis of a toric ideal cannot be bounded by a polynomial function of the true degrees of its circuits, highlighting a fundamental complexity difference.

Contribution

It establishes that Graver degrees are not polynomially bounded by circuit degrees in toric ideals, revealing a key limitation in their relationship.

Findings

Graver degrees are unbounded by polynomial functions of circuit degrees

The result impacts the understanding of toric ideal complexity

Provides a theoretical foundation for future algebraic complexity analysis

Abstract

Let $I_A$ be a toric ideal. We prove that the degrees of the elements of the Graver basis of $I_A$ are not polynomially bounded by the true degrees of the circuits of $I_A$.