  Let $f$ be a polynomial with coefficients in the ring $O_S$ of $S$-integers
of a number field $K$, $b$ a non-zero $S$-integer, and $m$ an integer $\ge 2$.
We consider the equation $( \star )$: $f(x) = b y^m$ in $x,y \in O_S$. Under
the well-known LeVeque condition, we give fully explicit upper bounds in terms
of $K, S, f, m$ and the $S$-norm of $b$ for the heights of the solutions $x$ of
the equation $( \star)$. Further, we give an explicit bound $C$ in terms of $K,
S, f$ and the $S$-norm of $b$ such that if $m > C$ the equation $(\star)$ has
only solutions with $y = 0$ or a root of unity. Our results are more detailed
versions of work of Trelina, Brindza, Shorey and Tijdeman, Voutier and Bugeaud,
and extend earlier results of B\'erczes, Evertse, and Gy\H{o}ry to polynomials
with multiple roots. In contrast with the previous results, our bounds depend
on the $S$-norm of $b$ instead of its height.
