Factoring translates of polynomials (Arvind Suresh, University of Arizona – Tucson)

Given a degree d polynomial f(x) in Q[x], consider the subset S_f  of Q consisting of rational numbers t for which the translated polynomial f(x) – t factors completely in Q[x]. For example, if f is linear or quadratic then S_f is always infinite, but if degree of f is at least 3, then interesting things can happen. In this talk, we discuss a connection between the set S_f and the classical Prouhet–Tarry–Escott problem (which asks for integer solutions to certain symmetric family of equations), and we present two infinite families of polynomials f for which S_f is infinite (upon replacing Q with certain number fields). Time permitting, we outline how these can then be used to produce algebraic curves over number fields having a record number of rational points (relative to their genus).