On the Gauss algebra of toric algebras

TL;DR

This paper investigates the structure and generators of the Gauss algebra associated with certain monomial-generated subalgebras of polynomial rings, providing explicit descriptions for specific classes like Borel fixed and Veronese algebras.

Contribution

It characterizes the generators and structure of the Gauss algebra for various classes of monomial algebras, including Borel fixed, squarefree Veronese, and edge rings of bipartite graphs.

Findings

Generators of $ ext{GG}(A)$ are monomials of degree $(r-1)d$.

Explicit descriptions of $ ext{GG}(A)$ for Borel fixed and Veronese algebras.

Embedding dimension of $ ext{GG}(A)$ for bipartite graphs with loops is bounded by graph complexity.

Abstract

Let $A$ be a $K$-subalgebra of the polynomial ring $S=K[x_1,\ldots,x_d]$ of dimension $d$, generated by finitely many monomials of degree $r$. Then the Gauss algebra $\GG(A)$ of $A$ is generated by monomials of degree $(r-1)d$ in $S$. We describe the generators and the structure of $\GG(A)$, when $A$ is a Borel fixed algebra, a squarefree Veronese algebra, generated in degree $2$, or the edge ring of a bipartite graph with at least one loop. For a bipartite graph $G$ with one loop, the embedding dimension of $\GG(A)$ is bounded by the complexity of the graph $G$.