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

Abstract

We prove that for any toric ideal of a graph the degree of any element of Graver basis is bounded above by an exponential function of the maximal degree of a circuit.

On the relative size of toric bases

“…In [13] it was proved that for any toric ideal I G of a graph G the degree of any element of the Graver basis of I G is bounded above by an exponential function of the maximal degree of a circuit. It is an interesting problem if this is true for any toric ideal.…”

“…There are several results in the literature concerning degree bounds of the elements of these sets and sometimes bounds on the one of these sets in terms of another set. There exist several bounds on the degrees of the elements of the Graver basis of a toric ideal which have important implications to integer programming and computational algebraic geometry, see for example [6,8,9,13,14,15,16,19].…”

The positivity of local equivariant Hirzebruch class for toric varieties

“…His main research interests include group actions and equivariant cohomology theories in algebraic geometry, geometric representation theory and generalised cohomology theories. He has published his BSc thesis [97] on toric ideals in a post-conference volume and the MSc thesis [98] on equivariant characteristic classes in Bulletin of London Mathematical Society. During the PhD, under the supervision of Tamás Hausel, Kamil has worked on the relations between cohomology theories and rings of functions on fixed point and zero schemes.…”

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 − .…”