Bombieri--Weyl Contractivity and Rigidity for Homogeneous Polynomials
Abstract
We determine the dimension-free threshold for the comparison between the Bombieri--Weyl norm and the supremum norm of complex homogeneous polynomials on ℓpn. For m-homogeneous polynomials, the critical scale is p=2m: below this threshold no dimension-free comparison is possible, while at p=2m we obtain |P|BW≤ (1+C/m)|P|2m. Moreover, there exist absolute constants A>0 and m0∈ℕ such that, for every m≥ m0 and p≥ 2m+A, |P|BW≤|P|p, with optimal constant one. Equality holds precisely for coordinate pure powers. We also obtain a quantitative stability statement for near-extremizers. The proof is based on a decomposition by multiplicity patterns, contractive orbit projections, Hardy--Littlewood estimates for reduced multilinear forms, and Wiener-type slice estimates. Finally, we show that the corresponding contractive phenomenon fails over the real scalar field.
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BibTeX
@misc{nunezalarcon2026bombieriweyl,
title = {{Bombieri--Weyl Contractivity and Rigidity for Homogeneous Polynomials}},
author = {Daniel Nunez-Alarcon and Daniel M. Pellegrino and Anselmo Raposo Jr. and Eduardo V. Teixeira},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.13753},
eprint = {2608.13753},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.13753},
}
RIS
TY - UNPB
TI - Bombieri--Weyl Contractivity and Rigidity for Homogeneous Polynomials
AU - Daniel Nunez-Alarcon
AU - Daniel M. Pellegrino
AU - Anselmo Raposo Jr.
AU - Eduardo V. Teixeira
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2608.13753
UR - https://arxiv.org/abs/2608.13753
AB - We determine the dimension-free threshold for the comparison between the Bombieri--Weyl norm and the supremum norm of complex homogeneous polynomials on ℓ_p^n. For m-homogeneous polynomials, the critical scale is p=2m: below this threshold no dimension-free comparison is possible, while at p=2m we obtain |P|_BW≤ (1+C/m)|P|_2m. Moreover, there exist absolute constants A>0 and m_0∈ℕ such that, for every m≥ m_0 and p≥ 2m+A, |P|_BW≤|P|_p, with optimal constant one. Equality holds precisely for coordinate pure powers. We also obtain a quantitative stability statement for near-extremizers. The proof is based on a decomposition by multiplicity patterns, contractive orbit projections, Hardy--Littlewood estimates for reduced multilinear forms, and Wiener-type slice estimates. Finally, we show that the corresponding contractive phenomenon fails over the real scalar field.
ER -