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

30 pages. This version supersedes the previous version. It contains new and stronger results, including Bombieri--Weyl contractivity and rigidity, and appears under a new title