The Orbit Intersection Problem and Polynomial Functions

The first part of this thesis considers the problem of reconstructing a rational function from one of its orbits. Conjecturally, two complex rational functions of degree greater than one possess orbits with infinite intersection if and only if they have a common iterate. This conjecture has been recently verified in the polynomial case, however the rational function case is vastly more difficult. Our first main theorem verifies this conjecture for rational functions of coprime degree, and is the first to address intersections of orbits of rational functions which are not polynomials. The second part of this thesis considers the problem of characterizing polynomial functions using only their values on natural numbers. Our second main theorem proves that if the $m$-th divided difference of a sequence $s_n$ of rational numbers is integer-valued, then the sequence is given by a polynomial in $n$ if and only if there is a positive number $theta$ with $s_n ll theta^n$ and $1+theta < e^{{1}+tfrac12+cdots+tfrac1m}$.PHDMathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/163231/1/aodesky_1.pd

Effective Methods for Diophantine Equations

UBL - phd migration 201