M.Sc Student M.Sc Thesis Segol Nimrod Injectiveness of Rational Functions Department of Mathematics ASSOCIATE PROF. Danny Neftin Abstract

The most difficult open case of the Carney Hortsch Zieve conjecture for curves is to show that for every number field K, there exists a constant N, such that for any rational function f with coefficients in K, the induces map P1 (K)-> P1 (K) that is at most N-to-1 outside of a finite set. With the exception of the case of polynomials which was solved by Carney Hortsch and Zieve, the previous work on this conjecture concerns only indecomposable maps f.

Let k be the algebraic closure of the rational numbers. Our main contribution is building on top of geometrically indecomposable maps that induce a map which is at least 2-to-1 for infinitely many valued in some number field. We show that if f=f2 composed with f1 induces a map that is at least 2-to-1 for infinitely many values in some number field, with f1 a function such that the numerator of (f1(X)-f1(Y))/(X-Y) is irreducible and the Galois closure of k(x)/k(f1(x)) has genus >1, then either f is in an explicit list of functions with bounded degree, or f2 is 2-to-1 or f=L composed with f0 where L a subLattès map and f0 of bounded degree. We show that if f=f2 composed with f1, induces a map that is at least 2-to-1 for infinitely many values in some number field, with f1 a geometrically indecomposable subLattès map, then either f is a subLattès map or f2 is 2-to-1.

We show that there exists a bound N depending only on the number field K, such that for all new examples of functions f that induce a map that is at least 2-to-1 for infinitely many values in K, the induces map f: P1 (K)-> P1 (K) is at most N-to-1 outside of a finite set.