Toric completions and bounded functions on real algebraic varieties

TL;DR

This paper explores how to compactify semi-algebraic sets using toric varieties, enabling a combinatorial understanding of polynomial growth, bounded functions, and applications to positive polynomials and the moment problem.

Contribution

It extends previous work to compute bounded functions in toric compactifications and demonstrates the absence of wild behaviors in this setting.

Findings

Computed the ring of bounded functions in toric compactifications.

Described asymptotic growth of polynomials via combinatorial data.

Showed certain pathological examples cannot occur in toric settings.

Abstract

Given a semi-algebraic set S, we study compactifications of S that arise from embeddings into complete toric varieties. This makes it possible to describe the asymptotic growth of polynomial functions on S in terms of combinatorial data. We extend our earlier work to compute the ring of bounded functions in this setting and discuss applications to positive polynomials and the moment problem. Complete results are obtained in special cases, like sets defined by binomial inequalities. We also show that the wild behaviour of certain examples constructed by Krug and by Mondal-Netzer cannot occur in a toric setting.