Functional Equations Involving Laurent Polynomials and Meromorphic Functions, with Applications to Dynamics and Diophantine Equations.

摘要

In this thesis, our main theorem gives the classification of all Laurent polynomials $f(X)$ such that the numerator of $frac{f(X)-f(Y)}{X-Y}$ has an irreducible factor whose normalization has genus zero or one. The work in this thesis uses various tools from algebraic geometry, and heavily relies on Galois theory and the classification of finite simple groups. As an application of our main theorem, we prove a theorem which gives all the solutions to the functional equation $fcirc P=fcirc Q$, where $f$ is a complex Laurent polynomial and $P, Q$ are distinct complex meromorphic functions. This theorem gives many classes of negative examples to an open question of Lyubich and Minsky. Moreover, this theorem has consequences for many important problems in complex dynamics and the distribution of values of meromorphic functions, since these problems can be reduced to solving the functional equation.As another application of the main theorem, we prove a theorem which gives all Laurent polynomials, such that there are infinitely many $c$ in a number field for which $f(X)=c$ has at least two solutions in the number field. The polynomial case analogue was recently proved by Carney, Hortsch and Zieve, and they used the analogue to prove the following unexpected result: for any polynomial $f(X)inmathbb{Q}[X]$, the function $f: mathbb{Q}rightarrowmathbb{Q}$ defined by $xmapsto f(x)$ is at most $6$-to-$1$ for all but finitely many values. A Laurent polynomial analogue of their result can be expected using the theorem in this thesis. It will provide evidence in support of an analogous conjecture about rational functions which would be a far-reaching generalization of the results of Mazur and Merel about rational torsion points on elliptic curves.