Toric completions and bounded functions on real algebraic varieties (Q2835338)

In the paper under review, the authors extend their work from [Trans. Am. Math. Soc. 364, No. 9, 4663--4682 (2012; Zbl 1279.14072)]. Given a normal affine real varity \(V\) and a semi-algebraic subset \(S\) of \(V\), the idea of that paper was to analyze the ring \(B_V(S)\) of regular functions on \(V\) which are bounded on \(S\) by a completion of \(V\) that have been called \(S\)-compatible. Here are the definitions: A \textit{completion} of \(V\) is an open dense embedding \(V\hookrightarrow X\) into a normal complete real variety. The completion \(X\) is said to be \textit{\(S\)-compatible} if, for every irreducible component \(Z\) of \(X\setminus V\), the set \(Z(\mathbb{R})\cap\overline{S}\) is either empty or Zariski-dense in \(Z\). In the present paper the authors use toric completions: Let \(V\) be an affine toric varity. A \textit{toric completion} of \(V\) is an open embedding of \(V\) into a complete toric varity \(X\) which is compatible with the torus actions. Given a semi-algebraic subset \(S\) of \(V\), the existence of a toric \(S\)-compatible completion allows to describe the asymptotic growth of polynomial functions on \(S\) in terms of combinatorial data. In particular, the ring \(B_V(S)\) can be made explicitely in this case.