Alexander Zvonkin (LaBRI, Université de Bordeaux)
zvonkin@labri.fr
Half a century ago, in 1965, Birch, Chowla, Hall, and Schinzel posed the following question. Let $A$ and $B$ be two coprime polynomials. What is the minimum possible degree of the difference $A^3 - B^2$? In 1995, Zannier considered a more general problem: what is the minimum degree of the difference of two polynomials with a prescribed factorization pattern? Subsequent studies revealed two phenomena. First, when the minimum degree is attained, the polynomials in question turn out to be defined over number fields; therefore, the universal Galois group (the automorphism group of the field of algebraic numbers) acts on these polynomials. Second, the theory of dessins d'enfants makes it clear that the problem is closely related to the study of "weighted trees". These are plane trees whose edges are endowed with positive integral weights. In particular, the Galois group acts also on such trees, and many aspects of this action, both on trees and on polynomials, can be explained by combinatorial properties of the trees.
This is a joint work with Fedor Pakovich and Nikolay Adrianov.