|M.Sc Student||Segol Nimrod|
|Subject||Injectiveness of Rational Functions|
|Department||Department of Mathematics||Supervisor||Dr. Danny Neftin|
|Full Thesis text|
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.