Graver degrees are not polynomially bounded by true circuit degrees

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 bounded above by a polynomial on the maximal true degree of the circuits of I A .

A Bound on Degrees of Primitive Elements of Toric Ideals of Graphs

“…We provided the positive answer for toric graph ideals only. In [8] toric graph ideals were used as counterexamples to polynomial bounds. It was able because of the pictorial description of their minimal binomials, given in [10] and [5].…”

“…, a m ). In [9] Tatakis and Thoma disproved the conjecture and in [8] they proved that there is no polynomial bound. They provide counterexamples of toric graph ideals for which the Graver degrees are exponentialy large compared to the true circuit degrees.…”

“…The search for counterexamples among graph ideals fails this time and we prove that for toric graph ideals the Graver basis degrees are bounded by an exponential function of maximal true circuit degrees. It's now known that for toric graph ideals the true degree of a circuit is equal to its usual degree (see Theorem 3.1 in [8]), thus we have to bound the degrees of primitive elements by an exponential function of the maximal usual degree of a circuit. This is done in Theorem 2.5.…”

On toric ideals arising from signed graphs

“…Among them, the set of primitive binomials, which is known to form the Graver basis, was studied widely related to a problem initiated by Sturmfels, called true degree problem. (See [23,25,26,28] for detail.) For a toric ideal I A , an irreducible binomial B = B + − B − of I A is primitive if there exists no other binomial B 0 = B + 0 − B − 0 such that B + 0 |B + and B − 0 |B − .…”

On toric ideals arising from signed graphs

“…Among them, the set of primitive binomials, which is known to form the Graver basis, was studied widely related to a problem initiated by Sturmfels, called true degree problem. (See [22,24,25,27] for detail.) For a toric ideal I A , an irreducible binomial B " B `´B ´of I A is primitive if there exists no other binomial B 0 " B 0 ´B0 such that B 0 |B `and B 0 |B ´.…”