Quadratic Generated Normal Domains From Graphs

TL;DR

This paper characterizes when quadratic-monomial generated subrings of Laurent polynomial rings are normal, using a combinatorial graph structure and a generalized odd cycle condition.

Contribution

It provides a complete characterization of normality and normalizations for quadratic-monomial generated domains via a novel combinatorial graph approach.

Findings

Normality characterized by a generalized odd cycle condition on a mixed signed, directed graph.

Developed a combinatorial structure linking quadratic monomials to graph edges.

Classified normalizations of quadratic-monomial generated domains.

Abstract

Determining whether an arbitrary subring $R$ of $k[x_1^{\pm 1},\dots, x_n^{\pm 1}]$ is a normal domain is, in general, a nontrivial problem, even in the special case of a monomial generated domain. In this paper, we provide a complete characterization of the normality and normalizations of quadratic-monomial generated domains. For a quadratic-monomial generated domain $R$, we develop a combinatorial structure that assigns, to each quadratic monomial of the ring, an edge in a mixed signed, directed graph $G$, i.e., a graph with signed edges and directed edges. We classify the normality and the normalizations of such rings in terms of a generalization of the combinatorial odd cycle condition on $G$.