Seminar

I shall begin by describing the solution (by Bombieri and Pila) to the following problem of Sarnak: prove that if f is a real analytic, but not algebraic, function defined on the closed interval [0, 1], then the equation f(s)=q has rather few solutions in rational numbers.

Once the terms here have been made precise, one can then formulate a natural conjecture for analytic functions f of several variables. It turns out that the number-theoretic part of the argument in the one variable case generalises easily. However, the difficulty comes in the analysis, and it is here that techniques from model theory, specifically o-minimality, play a role.