Homogeneous Khovanskii bases and MUVAK bases

TL;DR

This paper extends the concept of Khovanskii bases to include arbitrary gradings and multiple valuations, introducing homogeneous and MUVAK bases with algorithms for their computation, relevant for algebraic geometry applications.

Contribution

It generalizes Khovanskii bases by defining homogeneous and MUVAK bases, providing algorithms for their computation, and improving existing algorithms in specific geometric contexts.

Findings

Introduction of homogeneous Khovanskii bases respecting arbitrary gradings.

Definition and computation algorithms for MUVAK bases with multiple valuations.

Application to computing Cox rings of minimal models of quotient singularities.

Abstract

In 2019, Kaveh and Manon introduced Khovanskii bases as a special 'Gr\"obner-like' generating system of an algebra. We extend their work by considering an arbitrary grading on the algebra and propose a definition for a 'homogeneous Khovanskii basis' that respects this grading. We generalize Khovanskii bases further by taking multiple valuations into account (MUVAK bases). We give algorithms in both cases. MUVAK bases appear in the computation of the Cox ring of a minimal model of a quotient singularity. Our algorithm is an improvement of an algorithm by Yamagishi in this situation.