Sponsored by the Rutgers University Department of Mathematics and the

Center for Discrete Mathematics and Theoretical Computer Science (DIMACS)

**Co-organizers:****Drew Sills**, Rutgers University, asills {at} math [dot] rutgers [dot] edu g**Doron Zeilberger**, Rutgers University, zeilberg {at} math [dot] rutgers [dot] edu

Title: Permutation Polynomials

Speaker: **Michael Zieve**, Rutgers University

Date: Thursday, April 12, 2007 5:00pm

Location: Hill Center, Room 705, Rutgers University, Busch Campus, Piscataway, NJ

Abstract:

A polynomial f(x) is a permutation polynomial (PP) over a field k if the map a-->f(a) permutes the elements of k. I will present various results about PPs: for instance, it is known that any PP over GF(q) of degree less than q^(1/4) will be a PP over GF(q^n) for infinitely many n. Thanks to the combined efforts of several mathematicians, we now have a conjecturally complete list of PPs of degree less than q^(1/4), some of which are quite complicated. My talk will be accessible to all, but I will touch on the methods used: the classification of finite simple groups, the Riemann hypothesis for curves over finite fields, Galois cohomology, higher ramification groups, bounds on automorphism groups of curves, etc.