Squarefree parts of polynomial values

Abstract

Given a separable nonconstant polynomial f(x) with integer coefficients, we consider the set S consisting of the squarefree parts of all the rational values of f(x), and study its behavior modulo primes. Fixing a prime p, we determine necessary and sufficient conditions for S to contain an element divisible by p. We conjecture that if p is large enough, then S contains infinitely many representatives from every nonzero residue class modulo p. The conjecture is proved by elementary means assuming f(x) has degree 1 or 2. If f(x) has degree 3, or if it has degree 4 and has a rational root, the conjecture is shown to follow from the parity conjecture for elliptic curves. For polynomials of arbitrary degree, a local analogue of the conjecture is proved using standard results from class field theory, and empirical evidence is given to support the global version of the conjecture.