2012/11/28 by Rudnick, Zeev
#11N32 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1211.6733
We study representation of square-free polynomials in the polynomial ring F[t] over a finite field F by polynomials in F[t][x]. This is a function field version of the well-studied problem of representing squarefree integers by integer polynomials, where it is conjectured that a separable polynomial f(x) with integer coefficients takes infinitely many squarefree values, barring some simple exceptional cases, in fact that the integers n for which f(n) is squarefree have a positive density. We show that if f(x) in F[t][x] is separable, with square-free content, of bounded degree and height, then as the finite field size #F tends to infinity, for almost all monic polynomials a(t), the polynomial f(a) is squarefree.