On generalized cyclotomic derivations

TL;DR

This paper investigates the properties of generalized cyclotomic derivations, focusing on their rational constants and Darboux polynomials, establishing conditions for the absence of Darboux polynomials and exploring tensor product cases.

Contribution

It provides a characterization of when a generalized cyclotomic derivation has no Darboux polynomials, linking it to the field of rational constants, and extends results to tensor products.

Findings

Derivation has no Darboux polynomials iff its field of rational constants equals the base field

Characterization of Darboux polynomials for generalized cyclotomic derivations

Extension of results to tensor products of polynomial algebras

Abstract

In this article we study the field of rational constants and Darboux polynomials of a generalized cyclotomic $K$-derivation $d$ of $K[X]$. It is shown that $d$ is without Darboux polynomials if and only if $K(X)^d=K$. Result is also studied in the tensor product of polynomial algebras.