The optimal multilinear Bohnenblust-Hille constants: a computational solution for the real case

TL;DR

This paper introduces a computational method using Wolfram Language to determine optimal multilinear Bohnenblust-Hille constants, advancing understanding of their behavior and testing related conjectures.

Contribution

It provides a computational approach to find optimal constants for the Bohnenblust-Hille inequality, improving upon previous algorithms and exploring new geometric insights.

Findings

Confirmed the conjecture $C_m=2^{1-1/m}$ for certain m

Discovered new geometric properties of $B_{\mathcal{L}(^3\mathbb{R}^3)}$

Demonstrated effectiveness of Wolfram Language in complex mathematical computations

Abstract

The Bohnenblust-Hille inequality for $m$-linear forms was proven in 1931 as a generalization of the famous 4/3-Littlewood inequality. The optimal constants (or at least their asymptotic behavior as $m$ grows) is unknown, but significant for applications. A recent result, obtained by Cavalcante, Teixeira and Pellegrino, provides a kind of algorithm, composed by finitely many elementary steps, giving as the final outcome the optimal truncated Bohnenblust-Hille constants of any order. But the procedure of Cavalcante \textit{et al.} has a fairly large number of calculations and computer assistance cannot be avoided. In this paper we present a computational solution to the problem, using the Wolfram Language. We also use this approach to investigate a conjecture raised by Pellegrino and Teixeira, asserting that $C_m=2^{1-1/m}$ for all $m\in\mathbb{N}$ and to reveal interesting unknown facts about the geometry of $B_{\mathcal{L}(^3\mathbb{R}^3)}$.