Real-Rooted Polynomials, Entropic Inequalities, and First Proof

Abstract

A sum of independent random variables satisfies a number of information theoretic inequalities, such as the entropy power inequality and Stam's inequality. I will explain how analogous results hold in the context of real-rooted polynomials, where the entropy is replaced by the log discriminant of the polynomial, the convolution is replaced by the finite free convolution, and Gaussians are replaced by Hermite polynomials. These results can be interpreted as quantitative statements about how the spacings of the zeros of a polynomial evolve under differential operators that preserve real-rootedness. Their proofs combine ideas from classical and free probability and the theory of hyperbolic polynomials.

One of these theorems --- the finite free Stam inequality --- also appeared as Problem 4 in the first batch of the First Proof project, which was designed to assess how well AI systems can solve naturally arising, previously unpublished research questions. I will use it as a case study in a broader discussion of what it means to evaluate the capabilities of AI in research mathematics.

Joint work with J. Garza-Vargas and Z. Stier.

Nikhil Srivastava - Bio

Nikhil is a Professor of Mathematics at UC Berkeley and a Senior Scientist at the Simons Institute for the Theory of Computing. He received his PhD in Computer Science from Yale in 2010, advised by Daniel Spielman, and his BS from Union College in 2005. After postdocs at the IAS, Princeton, and MSRI, he spent two and a half years as a researcher at Microsoft Research India, before moving to Berkeley in 2015. He is a co-winner of the SIAM Polya Prize, the NAS Held Prize, the AMS Foias Prize, and the ICBS Frontiers of Science Award. More recently, he is on the editorial board of First Proof.