The complex polynomial Bohnenblust--Hille inequality states that, for a $d$-homogeneous polynomial $P:\mathbb C^N\to\mathbb C$, $P(z)=\sum_{|\alpha|=d}c_\alpha z^\alpha$, the coefficient norm $(\sum_{|\alpha|=d}|c_\alpha|^{p_d})^{1/p_d}$ is bounded by $B_d^{\mathrm{pol}}\|P\|_\infty$, where $p_d=2d/(d+1)$ and $B_d^{\ma...
A. Barbosa, D. Núñez-Alarcón, D. Pellegrino et al.· 0 citations
We study sparse Kahane--Salem--Zygmund constructions and weighted Hardy--Littlewood inequalities for homogeneous polynomials. For supports of cardinality $n^{d+o(1)}$, we determine the sharp power of $n$ governing the smallest norm of a unimodular $m$-linear form on $\ell_{p_1}^n\times\cdots\times\ell_{p_m}^n$; in the...
A. Barbosa, D. Núñez-Alarcón, A. Raposo et al.· 0 citations
The Kahane--Salem--Zygmund inequality provides unimodular $m$-linear forms with small supremum norm. We consider its dimension-free symmetric constant $C_m^{\mathrm{sym}}$ over the real scalar field, defined by the requirement that, for every $n$, some real symmetric unimodular $m$-linear form $A:(\ell_\infty^n)^m\to\m...
D. Pellegrino, A. Raposo, Eduardo V. Teixeira· 0 citations
We study real multilinear forms with coefficients in $\{-1,1\}$ on finite-dimensional Hilbert spaces. Every trilinear sign form on $\ell_2^r\times\ell_2^n\times\ell_2^n$ has norm at least $\sqrt n$. Writing $K_{r,n}$ for the least norm divided by $\sqrt n$, we prove that $K_{r,n}=1$ exactly when a Hadamard matrix of or...
Let $C_q$ denote the group of the $q$th roots of unity. A question arising from the work of Becker, Klein, Slote, Volberg and Zhang is whether the dimension-free Bohnenblust--Hille constants for functions on $C_q^N$ grow subexponentially with the degree. We answer this question affirmatively. In fact, we prove a strong...
We determine the exact finite-dimensional threshold in the zero-subspace problem of Aron and Rueda for complex homogeneous polynomials. More precisely, for every $d$ and $k$ we determine the least $m$ such that every $d$-homogeneous polynomial on $\mathbb{C}^m$ vanishes on a $k$-dimensional linear subspace. We also det...
N. Albuquerque, D. Pellegrino, A. Raposo· 0 citations
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