A new class of finitely generated polynomial subalgebras without finite SAGBI bases

TL;DR

This paper introduces a new class of finitely generated polynomial subalgebras that lack finite SAGBI bases, highlighting cases with infinitely many or continuum initial algebras, thus advancing understanding of subalgebra finite generation.

Contribution

It presents a novel class of finitely generated subalgebras with non-finitely generated initial algebras, including examples with continuum and finitely many initial algebras.

Findings

Existence of finitely generated subalgebras with non-finitely generated initial algebras

Construction of subalgebras with continuum many initial algebras

Identification of subalgebras with finitely many initial algebras

Abstract

The notion of initial ideal for an ideal of a polynomial ring appears in the theory of Gr\"obner basis. Similarly to the initial ideals, we can define the initial algebra for a subalgebra of a polynomial ring, or more generally of a Laurent polynomial ring, which is used in the theory of SAGBI (Subalgebra Analogue to Gr\"obner Bases for Ideals) basis. The initial algebra of a finitely generated subalgebra is not always finitely generated, and no general criterion for finite generation is known. The aim of this paper is to present a new class of finitely generated subalgebras having non-finitely generated initial algebras. The class contains a subalgebra for which the set of initial algebras is continuum, as well as a subalgebra with finitely many distinct initial algebras.