Finite Free Information Inequalities

TL;DR

This paper introduces finite free information theory for real-rooted polynomials, establishing analogues of key inequalities and resolving conjectures, with implications for free probability and potential theory.

Contribution

It develops finite free analogues of entropy and Fisher information inequalities, connecting real-rooted polynomial zeros with potential-theoretic inequalities and proving conjectures in the field.

Findings

Established finite free entropy and Fisher information inequalities.

Proved finite free analogues of Stam and entropy power inequalities.

Resolved conjectures by Shlyakhtenko and Gribinski.

Abstract

We develop finite free information theory for real-rooted polynomials, establishing finite free analogues of entropy and Fisher information monotonicity, as well as the Stam and entropy power inequalities. These results resolve conjectures by Shlyakhtenko and Gribinski and recover inequalities in free probability in the large-degree limit. Equivalently, our results may be interpreted as potential-theoretic inequalities for the zeros of real-rooted polynomials under differential operators which preserve real-rootedness. Our proofs leverage a new connection between score vectors and Jacobians of root maps, combined with convexity results for hyperbolic polynomials.