Time: 9h, 31/7/2014

Location: Room 4, Building A14, Institute of Mathematics, 18 Hoang Quoc Viet, Cau Giay, Hanoi

Abstract: Nevanlinna’s Five-Value theorem states that any two nonconstant meromorphic functions which have the same preimages as one another at five different points are identical.  Several authors have sought variants of this result which leads to the study of the functional equation  P o F = Q o G  where P and Q are polynomials and F and G are nonconstant meromorphic functions.  We will present all solutions to this equation.  In addition, we will describe all polynomials P and Q with algebraic coefficients for which the equation P(X)=Q(Y) has infinitely many solutions in some algebraic number field.  Our proofs rely on tools from complex analysis, group theory, Galois theory, algebraic geometry, representation theory, algebraic topology, differential equations, combinatorics, number theory, and other subjects.
In this talk I will present the proof of the irreducible case- when P(X)=Q(Y) is irreducible of genus 0 or 1, which can be reduced to a pure combinatorial problem by Riemann-Hurwitz formula.