Transcendence proofs using Mahler functions

par Enzo Brechler

lundi 8 juin 2026, 10:30 → 11:30 Europe/Paris

Fokko du Cloux (ICJ)

Proving that a given real number is transcendental is in general a difficult problem. In 1929, Mahler introduced a new method to prove that many numbers such as x=0.1101000100000001... are transcendental, giving rise to the class of so-called Mahler functions. These are analytic functions with algebraic coefficients which satisfy a linear difference equation associated to the Mahler operator z->z^q. The theory of Mahler functions has greatly evolved, to the point that we now have a way to systematically determine if f(alpha) is algebraic or transcendental when f is a Mahler function and alpha is an algebraic number. One of the main examples of numbers that can be obtained in this way are the automatic numbers such as the Thue-Morse number x=0.110100110010110... In this talk, I will present the main results on Mahler functions and how to apply them concretely to get transcendence results. Furthermore, I will present a generalisation of the theory to the case of Mahler functions of multiple variables, and show that it gives transcendence results for so-called morphic numbers (an example being the number derived from the Fibonacci word x=0.1001010010..., which is obtained by iterating the substitution rule 0->01, 1->0).