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

TL;DR

This paper studies closed and factorially closed polynomials in multiple variables, characterizes them over algebraically closed fields, and applies these results to determine kernels of monomial derivations in polynomial rings.

Contribution

It provides a new characterization of factorially closed polynomials and applies this to explicitly determine kernels of monomial derivations.

Findings

Characterization of factorially closed polynomials in n variables

Determination of kernels of non-zero monomial derivations in two variables

Identification of monomial derivations with specific kernel properties

Abstract

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 $k(x,y)$.