Finite Free Information Inequalities

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.