Khovanskii-Finite Valuations, Rational Curves, and Torus Actions

TL;DR

This paper investigates conditions under which valuations on graded domains have finite Khovanskii bases, focusing on rational curves and torus actions, and reveals limitations of existing tropicalization procedures.

Contribution

It introduces a reduction from multigraded to simply graded domains and connects to problems by Kaveh and Manon, showing certain tropical procedures may not terminate.

Findings

Reduction from multigraded to simply graded domains

Finite Khovanskii bases for rational curves and almost toric varieties

Tropical prime cone procedures may not terminate in general

Abstract

We study full rank homogeneous valuations on (multi)-graded domains and ask when they have finite Khovanskii bases. We show that there is a natural reduction from multigraded to simply graded domains. As special cases, we consider projective coordinate rings of rational curves, and almost toric varieties. Our results relate to several problems posed by Kaveh and Manon, and imply that the procedure of Bossinger-Lamboglia-Mincheva-Mohammadi for producing tropical prime cones will not terminate in general.