Square-free values of polynomials over the rational function field

Abstract

We study representation of square-free polynomials in the polynomial ring Fq[t] over a finite field Fq by polynomials in Fq[t][x]. This is a function field version of the well-studied problem of representing square-free integers by integer polynomials, where it is conjectured that a separable polynomial f∈Z[x] takes infinitely many square-free values, barring some simple exceptional cases, in fact that the integers a for which f(a) is square-free have a positive density. We show that if f(x)∈Fq[t][x] is separable, with square-free content, of bounded degree and height, and n is fixed, then as q→∞, for almost all monic polynomials a(t) of degree n, the polynomial f(a) is square-free.