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Polynomial growth of complex polynomial Bohnenblust--Hille constants
Abstract
For complex m-homogeneous polynomials, let Dm denote the optimal dimension-free constant in the polynomial Bohnenblust--Hille inequality. We prove that, for every B>1/2, there exists KB>0 such that Dm≤ KB mB, m≥1. More precisely, we obtain the square-root-scale estimate Dm≤ √m exp (C√(log m) loglog m). The proof combines coefficient-preserving degree reduction with a weighted bootstrap, separating balanced and dominant degree profiles. We also determine the critical linear-dimensional asymptotics: Dm,nm⟶ 2 whenever nm/m⟶1, and consequently liminfm→∞Dm≥2. Further applications give two-sided bounds for homogeneous Sidon constants and a quantitative logarithmic remainder in the multidimensional Bohr-radius asymptotic.
Related
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- Rigidity of Exact Contractivity in the Complex Polynomial Bohnenblust--Hille Inequality
- Asymptotic Contractivity of Bohnenblust-Hille Constants with Bounded Monomial Support
- The Geometry of Real Anisotropic Bohnenblust--Hille Constants
- Polynomial Bohnenblust--Hille bounds for product of cyclic groups
BibTeX
@misc{pellegrino2026polynomial,
title = {{Polynomial growth of complex polynomial Bohnenblust--Hille constants}},
author = {Daniel M. Pellegrino and Eduardo V. Teixeira},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.16584},
eprint = {2608.16584},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.16584},
}
RIS
TY - UNPB
TI - Polynomial growth of complex polynomial Bohnenblust--Hille constants
AU - Daniel M. Pellegrino
AU - Eduardo V. Teixeira
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2608.16584
UR - https://arxiv.org/abs/2608.16584
AB - For complex m-homogeneous polynomials, let D_m denote the optimal dimension-free constant in the polynomial Bohnenblust--Hille inequality. We prove that, for every B>1/2, there exists K_B>0 such that D_m≤ K_B m^B, m≥1. More precisely, we obtain the square-root-scale estimate D_m≤ √m exp (C√(log m) loglog m). The proof combines coefficient-preserving degree reduction with a weighted bootstrap, separating balanced and dominant degree profiles. We also determine the critical linear-dimensional asymptotics: D_m,n_m⟶ 2 whenever n_m/m⟶1, and consequently liminf_m→∞D_m≥2. Further applications give two-sided bounds for homogeneous Sidon constants and a quantitative logarithmic remainder in the multidimensional Bohr-radius asymptotic.
ER -