PreprintCC BY 4.0arXiv
A Kallin's lemma for polynomials with exponents in a compact convex set and local approximation
Abstract
Let S⊂Bbb R+n be compact and convex with 0∈ S, and let 𝒫S be the algebra spanned by monomials with exponents in ⋃m≥0(mS∩Bbb N0n). We define the associated polynomial hull and prove a Kallin lemma for the union of two sets convex with respect to this algebra. A weighted approximation lemma then shows that, under a sector or ray condition, two admissible monomials generate exactly the functions of the unrestricted algebra that are constant on a specified zero fibre. Applications include an exact density criterion for mixed monomials, a transfer principle for relative perturbations, and density and hull computations on stars of totally real discs.
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BibTeX
@misc{chi2026kallins,
title = {{A Kallin's lemma for polynomials with exponents in a compact convex set and local approximation}},
author = {Kieu Phuong Chi and Duong Thi Thuy Van},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.32025},
eprint = {2610.32025},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.32025},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Kallin's lemma for polynomials with exponents in a compact convex set and local approximation
AU - Kieu Phuong Chi
AU - Duong Thi Thuy Van
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.32025
UR - https://arxiv.org/abs/2610.32025
AB - Let S⊂Bbb R_+^n be compact and convex with 0∈ S, and let 𝒫^S be the algebra spanned by monomials with exponents in ⋃_m≥0(mS∩Bbb N_0^n). We define the associated polynomial hull and prove a Kallin lemma for the union of two sets convex with respect to this algebra. A weighted approximation lemma then shows that, under a sector or ray condition, two admissible monomials generate exactly the functions of the unrestricted algebra that are constant on a specified zero fibre. Applications include an exact density criterion for mixed monomials, a transfer principle for relative perturbations, and density and hull computations on stars of totally real discs.
ER -