On the Gauss algebra of toric algebras

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 $G$.Comment: Accepted for publication in Journal of Algebraic Combinatoric