Toric completions and bounded functions on real algebraic varieties

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 in Plaumann and Scheiderer [‘The ring of bounded polynomials on a semi-algebraic set’, Trans. Amer. Math. Soc. 364 (2012) 4663–4682] 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 [‘Geometric interpretations of a counterexample to Hilbert's 14th problem, and rings of bounded polynomials on semialgebraic sets’, Preprint, 2011, arXiv:1105.2029] and Mondal-Netzer [‘How fast do polynomials grow on semialgebraic sets?’, J. Algebra 413 (2014) 320–344] cannot occur in a toric setting.