Seminar on Complex Analysis and Complex Geometry - Spring 2015

MONDAY, February 23, 2015, 3pm-4pm, PGH 646

Title: How many values does a polynomial map miss?

Speaker: Daqing Wan, University of California, Irvine

Abstract: For a polynomial map f(x) from a field F to itself, we are interested in the size of the values that f misses, that is, the cardinality of F - f(F). For F = C (the complex numbers), if f misses one value, then f is a constant (this is the fundamental theorem of algebra). For F = C, if a holomorphic map f misses two values, then f is again a constant (this is Picard's little theorem). What about when f: F^n -> F^n is a polynomial vector map? When F is a finite field F_q of q elements, this problem becomes very interesting. There are extensive results and open problems available. For example, if a polynomial f of degree d>1 misses one value of F_q, then it must miss at least (q-1)/d values. In this lecture, we give a self-contained exposition of the main results and the open problems on the value set problem, and explain its link to different parts of mathematics.