Sharp values for the constants in the polynomial Bohnenblust-Hille inequality

TL;DR

This paper precisely determines the Bohnenblust-Hille constants for 2-homogeneous polynomials in two variables over complex and real fields, and provides lower bounds for higher degrees in the real case.

Contribution

It establishes exact values for the Bohnenblust-Hille constants in the complex and real cases for degree 2 polynomials, and offers lower bounds for higher degrees in the real case.

Findings

Exact complex constant: .5^{1/4}

Exact real constant for degree 2

Lower bounds for higher degree real polynomials

Abstract

In this paper we prove that the complex polynomial Bohnenblust-Hille constant for $2$-homogeneous polynomials in ${\mathbb C}^2$ is exactly $\sqrt[4]{\frac{3}{2}}$. We also give the exact value of the real polynomial Bohnenblust-Hille constant for $2$-homogeneous polynomials in ${\mathbb R}^2$. Finally, we provide lower estimates for the real polynomial Bohnenblust-Hille constant for polynomials in ${\mathbb R}^2$ of higher degrees.