# The Krzy\.z Conjecture and an Entropy Conjecture

## TL;DR

This paper links the Krzyz conjecture to an entropy minimization problem for polynomials with roots on the unit circle, proposing that certain entropy conditions imply the conjecture's validity.

## Contribution

It establishes a conditional connection between entropy minimization and the Krzyz conjecture, offering a new approach to its proof under generic hypotheses.

## Key findings

- Entropy minimization for polynomials with roots on the unit circle relates to the Krzyz conjecture.
- Conditional proof of the Krzyz conjecture based on entropy and extremum hypotheses.
- Provides a new perspective linking polynomial root distributions to complex analysis conjectures.

## Abstract

We show that if the minimum entropy for a polynomial with roots on the unit circle is attained by polynomials with equally spaced roots, then, under a generic hypothesis about the nature of the extremum, the Krzyz conjecture on the maximum modulus of the Taylor coefficients of a holomorphic function that maps the disk to the punctured disk is true.