Typical Bohnenblust--Hille Ratios

Source abstract

The polynomial Bohnenblust--Hille inequality controls the coefficient $\ell_{2m/(m+1)}$-norm of an $m$-homogeneous polynomial by its supremum norm, with a constant independent of the dimension. We study the associated Bohnenblust--Hille ratio from a probabilistic point of view, by placing normalized surface measure on the Euclidean coefficient sphere. Our results reveal a sharp contrast between the extremal behavior governing the classical Bohnenblust--Hille constants and the typical scale seen in coefficient directions. For arbitrary prescribed monomial supports, the ratio is eventually at most $1$ almost surely, and it tends to zero in spherical measure exactly when the number of monomials tends to infinity. On the full complex polynomial spaces we determine its typical asymptotic scale uniformly in the dimension; in the critical regime $n_m/m\to1$ this gives $2\sqrt2/\sqrt{m\log m}$, in contrast with the nonvanishing extremal scale. For real polynomials, the corresponding ratio tends to zero in spherical measure exactly when the number of variables tends to infinity.