Weakly Newton-nondegenerate binomial ideals
Abstract
An ideal I of a polynomial ring R=k[x1,…,xn] is called weakly Newton-nondegenerate, or weakly NND, if its integral closure I̅ is a monomial ideal. We study weak Newton nondegeneracy for the family of quadratic binomial ideals I=(x12+ε1xa1xb1, …, xn2+εnxanxbn), εi∈{±1}, ai≠ bi, over an algebraically closed field. We prove that I is weakly NND if and only if I̅=m2, 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⊆{1,…,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.
Related
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BibTeX
@misc{hibi2026weakly,
title = {{Weakly Newton-nondegenerate binomial ideals}},
author = {Takayuki Hibi and Vinh Anh Pham},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.29039},
eprint = {2608.29039},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29039},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Weakly Newton-nondegenerate binomial ideals
AU - Takayuki Hibi
AU - Vinh Anh Pham
PY - 2026
DA - 2026/08/29
PB - arXiv
DO - 10.48550/arxiv.2608.29039
UR - https://arxiv.org/abs/2608.29039
AB - An ideal I of a polynomial ring R=k[x_1,…,x_n] is called weakly Newton-nondegenerate, or weakly NND, if its integral closure I̅ is a monomial ideal. We study weak Newton nondegeneracy for the family of quadratic binomial ideals I=(x_1^2+ε_1x_a_1x_b_1, …, x_n^2+ε_nx_a_nx_b_n), ε_i∈{±1}, a_i≠ b_i, over an algebraically closed field. We prove that I is weakly NND if and only if I̅=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⊆{1,…,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.
ER -