Oka-Weil approximation theorem for polynomials with exponents in compact convex sets in ℝn
Abstract
Let S⊂Rn+ be a compact convex set with 0∈ S, and let cPS 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 cPS-convex subsets of (C*)n and asked whether it holds for all compact cPS-convex subsets of Cn. We answer this question affirmatively when the cone 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 cPS-convex set is equivalent to the compactness of its hull with respect to a finitely generated monomial subring of cPS, to a corona-type certificate, and to the existence of a bounded neighbourhood defined by finitely many inequalities |pj|<cj with pj∈cPS. 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 cPS-hulls and the corresponding rational approximation theorem.
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BibTeX
@misc{van2026okaweil,
title = {{Oka-Weil approximation theorem for polynomials with exponents in compact convex sets in $\mathbb R^n$}},
author = {Duong Thi Thuy Van and Kieu Phuong Chi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.21181},
eprint = {2610.21181},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.21181},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Oka-Weil approximation theorem for polynomials with exponents in compact convex sets in ℝ^n
AU - Duong Thi Thuy Van
AU - Kieu Phuong Chi
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.21181
UR - https://arxiv.org/abs/2610.21181
AB - Let S⊂R^n_+ be a compact convex set with 0∈ S, and let 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 cP^S-convex subsets of (C^*)^n and asked whether it holds for all compact cP^S-convex subsets of C^n. We answer this question affirmatively when the cone 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 cP^S-convex set is equivalent to the compactness of its hull with respect to a finitely generated monomial subring of cP^S, to a corona-type certificate, and to the existence of a bounded neighbourhood defined by finitely many inequalities |p_j|<c_j with p_j∈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 cP^S-hulls and the corresponding rational approximation theorem.
ER -