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In this paper, we give some results on closed polynomials and factoriallyclosed polynomial in $n$ variables. In particular, we give a characterizationof factorially closed polynomials in $n$ variables over an algebraically closedfield for any characteristic. Furthermore, as an application of results onclosed polynomials, we determine kernels of non-zero monomial derivations onthe 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 $k(x,y)$.

Closed polynomials and their applications for computations of kernels of monomial derivations