A uniform boundedness conjecture for rational functions
Professor Michael Zieve
2016-12-27 16:00-17:00
P^1(Q) induced by any f(x) in Q(x) is (<=N)-to-1 over all but finitely many points. The key point here is that N is independent of f. This conjecture is a vast generalization of the uniform boundedness of rational torsion on elliptic curves (Mazur's theorem). Work on this conjecture has led to progress on both Hurwitz's existence problem (describing the ramification types of rational functions) and Zariski's thesis problem (describing the monodromy groups of rational functions).-->
610
Speaker: Professor Michael Zieve, (University of Michigan)
Place and Time: MCM 610, Dec 27 (Tuesday) 4:00pm-5:00pm
Title: A uniform boundedness conjecture for rational functions
Abstract: I will explain the progress that has been made towards proving the following conjecture: there is an absolute constant N such that the function f : P^1(Q)-->P^1(Q) induced by any f(x) in Q(x) is (<=N)-to-1 over all but finitely many points. The key point here is that N is independent of f. This conjecture is a vast generalization of the uniform boundedness of rational torsion on elliptic curves (Mazur's theorem). Work on this conjecture has led to progress on both Hurwitz's existence problem (describing the ramification types of rational functions) and Zariski's thesis problem (describing the monodromy groups of rational functions).
