Equations mod p restricted to boxes and applications

10/01/2013 - 12:15pm
10/01/2013 - 1:10pm
Mei-Chu Chang (UC Riverside)

The general type of problem considered is to establish uniform bounds on the number of solutions of an equation f(x, y) = 0 (mod p) with x, y restricted to intervals in general position and f(x, y) an irreducible polynomial of some degree. Thus in some sense, this is a (mod p) counterpart of the work of Bombieri and Pila. We will survey some results and applications, and sketch a proof of a particular case of the following problem. Let f(x), g(x) in F_p[x] be polynomials and let [0,M] intersected with Z be an interval. Then under certain assumptions, cardinality of the intersection of images f([0,M]) and g([0;M]) is M^{1-e}.

Mudd Science Library 126, Pomona College