# Manchester Geometry Seminar 2015/2016

**Thursday 5 November 2015. Joint meeting with Algebra Seminar.** *The Frank Adams Room (Room 1.212), the Alan Turing Building. 3pm*
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Galois Theory of Weighted Trees

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.