Oka-Weil approximation theorem for polynomials with exponents in compact convex sets in ${\mathbb{R}}^n$

Let $S\subset {\mathbb{R}}_{+}^n$ be a compact convex set with $0\in S$, and let ${\text{cP}}^S$ be the ring of polynomials whose exponents of the $m$-th degree lie in the dilates $mS$. Sigurðardóttir proved an Oka–Weil theorem for compact ${\text{cP}}^S$-convex subsets of $({\text{C}}^{\ast }{)}^n$ and asked whether it holds for all compact ${\text{cP}}^S$-convex subsets of ${\text{C}}^n$. We answer this question affirmatively when the cone ${\mathbb{R}}_{+}S$ is rational polyhedral, for every $S$ when $n=2$, and for several classes of nonpolyhedral cones in all dimensions; in all these cases the compact sets may meet the coordinate hyperplanes. For arbitrary $S$ we prove that the Oka–Weil property of a compact ${\text{cP}}^S$-convex set is equivalent to the compactness of its hull with respect to a finitely generated monomial subring of ${\text{cP}}^S$, to a corona-type certificate, and to the existence of a bounded neighbourhood defined by finitely many inequalities $\mathrm{\mid }{p}_j\mathrm{\mid }<c_j$ with $p_j\in {\text{cP}}^S$. The proofs use monomial charts, the classical Oka–Weil theorem in an auxiliary affine space, and elementary Gelfand theory. As applications we obtain an Oka–Stolzenberg criterion for ${\text{cP}}^S$-hulls and the corresponding rational approximation theorem.