On the Gauss Algebra of Toric Algebras

Abstract

Let A be a K-subalgebra of the polynomial ring S=K[x₁,…,xd] of dimension d, generated by finitely many monomials of degree r. Then the Gauss algebra G(A) of A is generated by monomials of degree (r−1)d in S. We describe the generators and the structure of G(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 G(A) is bounded by the complexity of the graph G.