Speaker:  Avner Kiro (Weizmann Institute of Science)

Title:  Integer-valued polynomials under growth constraints

Abstract:  We study integer-valued polynomials that grow at most exponentially on the integers |𝑃(𝑛)| ≤ 𝐴^𝑛 , a question originally raised by Dimitrov. A result of Elkies and Speyer shows a sharp threshold: below a certain growth rate only finitely many such polynomials exist, while above it there are infinitely many. In joint work with Alon Nishry, we obtained quantitative asymptotic estimates for how many such polynomials of large degree exist at or above this threshold. Our approach uses convex-geometric volume estimates together with potential theory and orthogonal polynomials. We also extend the framework to allow different growth bounds on the positive and negative integers, where the threshold is again determined by the logarithmic capacity of an explicit planar set.