Weakly Newton-nondegenerate binomial ideals

Abstract

An ideal $I$ of a polynomial ring $R=k[x_1,\dots,x_n]$ is called weakly Newton-nondegenerate, or weakly NND, if its integral closure $\overline{I}$ is a monomial ideal. We study weak Newton nondegeneracy for the family of quadratic binomial ideals \[ I=(x_1^2+\epsilon_1x_{a_1}x_{b_1},\ \dots,\ x_n^2+\epsilon_nx_{a_n}x_{b_n}),\qquad \epsilon_i\in\{\pm1\},\ a_i\neq b_i, \] over an algebraically closed field. We prove that $I$ is weakly NND if and only if $\overline I=\mathfrak{m}^2$, if and only if the given generators form a regular sequence, and if and only if an explicit combinatorial condition on the pair (support pattern, sign pattern) holds: no nonempty subset $S\subseteq\{1,\dots,n\}$ is simultaneously closed for the support data and sign-trivial for the associated lattice of relations. The last equivalence rests on a solvability criterion for systems of monomial equations over a divisible abelian group, in the spirit of Eisenbud and Sturmfels. As an application, we classify all such ideals for $n=3$.