A note on the polynomial Bohnenblust-Hille inequality

TL;DR

This paper establishes new lower bounds for the optimal constants in the polynomial Bohnenblust-Hille inequality, addressing a gap in the understanding of these bounds for higher-degree polynomials.

Contribution

It provides the first nontrivial lower bounds for the constants in the Bohnenblust-Hille inequality for n-homogeneous polynomials with n > 2.

Findings

Established lower bounds for the Bohnenblust-Hille constants.

Extended understanding of the inequality's behavior for higher-degree polynomials.

Contributed to the theoretical foundation of polynomial inequalities.

Abstract

Recently, in paper published in the Annals of Mathematics, it was shown that the Bohnenblust-Hille inequality for (complex) homogeneous polynomials is hypercontractive. However, and to the best of our knowledge, there is no result providing (nontrivial) lower bounds for the optimal constants for n-homogeneous polynomials (n > 2). In this short note we provide lower bounds for these famous constants.