Proof of the Agler--McCarthy entropy conjecture

TL;DR

This paper proves a sharp entropy inequality for polynomials with zeros on the unit circle, advancing the first step toward resolving the Krzyż conjecture.

Contribution

It establishes the homogeneous entropy inequality for such polynomials and characterizes the cases of equality, completing a key step in the proposed program.

Findings

Proved the sharp homogeneous entropy inequality for polynomials with zeros on the unit circle.

Determined the equality cases for the entropy inequality.

Progressed toward resolving the Krzyż conjecture by completing the first step.

Abstract

In 2021, J.~Agler and J.~E. McCarthy proposed a two-step programme toward the celebrated Krzy\.z conjecture. The first step is to prove an entropy conjecture for polynomials whose zeros all lie on the unit circle; the second is to establish a full degree condition for extremal functions in the Krzy\.z conjecture. The purpose of this paper is to complete the first step. More precisely, we establish the sharp homogeneous entropy inequality for all non-constant polynomials with zeros on the unit circle and determine the equality cases.