Toric ideals and their circuits (Q357884)

Let \(A\) be a configuration in \(\mathbb{R}^d\) and \(I_A\) its toric ideal. We say that an irreducible binomial \(f\in I_A\) is a circuit of \(I_A\) if there is no binomial \(g\in I_A\) such that \(\mathrm{supp}(g)\subset \mathrm{supp}(f)\) and \(\mathrm{supp}(g)\neq \mathrm{supp}(f)\).NEWLINENEWLINEThe present paper studies toric ideals generated by circuits. The authors give a sufficient condition for toric ideals which have squarefree quadratic initial ideals to be generated by circuits, cf. Proposition 1.1. Some ideals that satisfy this condition are Veronese subrings, second Veronese subrings and configurations arising from root systems, see Theorem 1.3 and Corollary 1.5. The authors also study toric ideals of finite graphs and characterize the graphs whose toric ideals are generated by circuits \(u-v\) such that either \(u\) or \(v\) are squarefree, see Theorem 2.4 and Theorem 2.6.