Given a power series f(x) with rational coefficients, under what conditions can one guarantee that f(x) is the Taylor expansion of a rational function?

This series of four lectures will present several criteria addressing this question. We begin with Borel’s theorem: if f(x) has integer coefficients and extends meromorphically to a disc of radius greater than 1, then it is rational. We then discuss various generalisations, allowing, for instance, rational coefficients or more general domains of meromorphy.

These criteria have striking applications in number theory. Examples include Dwork’s p-adic proof of the rationality of the zeta function of a variety over a finite field, the Bézivin–Robba approach to the Hermite–Lindemann–Weierstrass theorem on the transcendence of exponentials of algebraic numbers, or the recent proof by Dimitrov of the Schinzel–Zassenhaus conjecture. A selection of these applications will be presented, along with some related open problems.

These lectures are suitable for graduate and advanced undergraduate students, and all mathematicians interested in number theory.

Prerequisites: proof-based undergraduate courses in algebra and analysis, both real and complex.

The Ukrainian host is Asem Abdelraouf, a Simons Postdoc in Mathematics at Kyiv School of Economics (KSE).

Image: Fragment of M.C. Escher’s Print Gallery, 1956

Javier Fresán is a professor and the head of number theory group at Sorbonne University. His research covers arithmetic geometry, in particular the connections between algebraic equations, topology, and number theory. His work strengthens the links between theoretical mathematics and applied aspects in cryptography and physics. Javier Fresán is the Editor-in-chief of Documents Mathématiques and Journal of the Institute of Mathematics of Jussieu, and a member of editorial boards of several other mathematician journals.

Photo: polytechnique.edu