Invited speaker to the International Conference CGI, 02-06 September 2024 (Combinatorics and Geometry in Ioannina). 

Title: Robust and generalized robust toric ideals which are generated by quadrics and MG-toric ideals of graphs.

(joint works with I.Garcia-Marco & I.Marquez-Corbella)

Abstract: On the first part of this talk, we study robust and generalized robust toric ideals of graphs which are generated by quadrics. A toric ideal is called robust if its universal Grobner basis is a minimal set of generators, while it is called generalized robust if its universal Grobner basis equals its universal Markov basis. For toric ideals of graphs, we characterize combinatorially the graphs giving rise to robust and to generalized robust toric ideals generated by quadratic binomials. 

On the second part of this talk, we introduce the notion of MG-ideals. An ideal I is called an MG-ideal if it is minimally generated by a reduced Grobner basis of I. Ohsugi and Hibi proved that a toric ideal I_G of a bipartite graph G is generated by quadrics if and only if I_G has a Grobner basis consisting of quadrics, i.e. all its minimal generators have degree two, if and only if the polynomial ring K[G] is Koszul. We will see that for a bipartite graph G such that all minimal generators of I_G have the same degree μ>1, the ideal I_G is an MG-ideal, while this is not true for non bipartite graphs.

Conference webpage: Combinatorics and Geometry in Ioannina