How fast do polynomials grow on semialgebraic sets?

TL;DR

This paper investigates how polynomials grow on semialgebraic sets by analyzing associated graded algebras, revealing conditions for finite generation and illustrating diverse growth behaviors with examples relevant to moment problems.

Contribution

It introduces a framework linking polynomial growth to algebraic properties of semialgebraic sets, including new results on finite generation and counterexamples.

Findings

Finite generation of the algebra implies degree-based growth control.

Counterexamples show no universal link between degree and growth.

New three-dimensional sets with non-finitely generated bounded polynomial algebra.

Abstract

We study the growth of polynomials on semialgebraic sets. For this purpose we associate a graded algebra to the set, and address all kinds of questions about finite generation. We show that for a certain class of sets, the algebra is finitely generated. This implies that the total degree of a polynomial determines its growth on the set, at least modulo bounded polynomials. We however also provide several counterexamples, where there is no connection between total degree and growth. In the plane, we give a complete answer to our questions for certain simple sets, and we provide a systematic construction for examples and counterexamples. Some of our counterexamples are of particular interest for the study of moment problems, since none of the existing methods seems to be able to decide the problem there. We finally also provide new three-dimensional sets, for which the algebra of bounded polynomials is not finitely generated.