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 stronger estimate for functions whose Fourier characters involve at most $d$ coordinates. If $\BHint{d}{q}$ is the optimal constant for this larger class, then, for every fixed $q\geq2$, \[ \BHint{d}{q}\leq \exp\left(c_q\sqrt{d\log d} +O_q\left(\sqrt{\frac d{\log d}}\log\log d\right)\right), \] where $c_2=2$ and $c_q=\sqrt{2q\log(q-1)/(q-2)}$ for $q\geq3$. As an application, we obtain two-sided estimates for the Bohr radius of the Fourier layer formed by characters involving exactly $d$ coordinates, and we determine its asymptotic behaviour in natural joint regimes of $d$ and $N$.
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 determine the exact threshold for arbitrary polynomials of degree at most $d$ to be constant on a $k$-dimensional linear subspace. The two thresholds are different. In the homogeneous case the exact threshold follows from Tevelev's theorem on isotropic subspaces and closedness of the incidence locus. In the bounded-degree case we first eliminate the linear homogeneous component by passing to its kernel; the remaining components, of degrees $2,\ldots,d$, form the system to which the Debarre--Manivel theorem is applied. For $k=2$ we give a separate proof using top Chern classes and Newton's inequalities.
N. Albuquerque, Daniel Pellegrino, A. Raposo· 0 citations