Algorithmic Methods for Computing Monomial Invariants of Abelian Groups

Abstract

Given some arbitrary polynomial ring, an invariant polynomial is a polynomial that is unchanged by the action of a group G. We investigate the ring of invariant polynomials under the action of some abelian groups with the goal of finding generators for this ring. When considering an abelian group, we can always find a basis such that the action is diagonal, so there exists monomial generators mi for the invariant ring. By Noether’s degree bound, the minimal set of generating monomials ⟨m1, . . . ,mk⟩ is finite and the degree of each generating monomial mi is less than |G|. Motivated by the previous work of Gandini and Derksen, we present new theoretical approaches to find invariants of Abelian groups. In combination with the theory, we can apply exact algorithmic approaches and deploy parallel programming methods to create various algorithms for computing the minimal generating set of invariants for the ring of invariants. Finally, we present the Gandini-Ratliff-Rizzolo algorithm, a seeded generation algorithm for computing invariants.