  In this paper, we give some results on closed polynomials and factorially
closed polynomial in $n$ variables. In particular, we give a characterization
of factorially closed polynomials in $n$ variables over an algebraically closed
field for any characteristic. Furthermore, as an application of results on
closed polynomials, we determine kernels of non-zero monomial derivations on
the polynomial ring in two variables over a UFD. Finally, by using this result,
for a field $k$, we determine the non-zero monomial derivations $D$ on $k[x,y]$
such that the quotient field of the kernel of $D$ is not equal to the kernel of
$D$ in $k(x,y)$.
