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 $\ell_p^n$. 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 $\|P\|_{\mathrm{BW}}\le (1+C/m)\|P\|_{2m}$. Moreover, there exist absolute constants $A>0$ and $m_0\in\mathbb N$ such that, for every $m\ge m_0$ and $p\ge 2m+A$, $\|P\|_{\mathrm{BW}}\le\|P\|_p$, 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.