  We investigate the density of square-free values of polynomials with large
coefficients over the rational function field $\mathbb{F}_q[t]$. Some
interesting questions answered as special cases of our results include the
density of square-free polynomials in short intervals, and an asymptotic for
the number of representations of a large polynomial $N$ as a sum of a small
$k$-th power and a square-free polynomial.
