  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.
