Optimal upper bound for the circle-group (polydisc) Bohnenblust–Hille constant

Determine the optimal upper bound, as a function of the degree d, for the best constant BH_{T,d} in the circle-group Bohnenblust–Hille inequality: for every n ≥ 1 and every analytic polynomial f on the n-dimensional polytorus T^n of degree at most d, establish the smallest C(d) such that ∥f∥_{2d/(d+1)} ≤ C(d) ∥f∥_{T^n} holds uniformly over n. Equivalently, ascertain the sharp growth rate in d of BH_{T,d}.

Background

The classical Bohnenblust–Hille inequality on the polytorus T^n asserts that for analytic polynomials of degree at most d, ∥f∥{2d/(d+1)} ≤ C(d) ∥f∥{T^n} with a constant independent of n. The best known bound satisfies BH_{T,d} ≤ C√(d log d), improving earlier exponential estimates.

Despite substantial progress, the exact dependence on d is unknown. Pinpointing the optimal bound would settle a central quantitative question in holomorphic analysis in many variables and directly impact topics such as Dirichlet series, Bohr’s radius, and hypercontractivity for polynomials on complex tori.