Constants of cyclotomic derivations

TL;DR

This paper investigates the polynomial and rational constants of cyclotomic derivations in polynomial rings over fields containing roots of unity, revealing conditions under which these constants form polynomial rings or rational function fields.

Contribution

It characterizes the polynomial and rational constants of cyclotomic derivations, establishing when the constants form polynomial rings and describing their structure in specific cases.

Findings

Field of constants of $d$ is a rational function field in $n - (n)$ variables.

Ring of constants of $d$ is polynomial iff $n$ is a prime power.

Ring of constants of $ ext{Δ}$ is always $k[v]$, with $v$ as the product of variables.

Abstract

Let $k[X]=k[x_0,...,x_{n-1}]$ and $k[Y]=k[y_0,...,y_{n-1}]$ be the polynomial rings in $n\geqslant 3$ variables over a field $k$ of characteristic zero containing the $n$-th roots of unity. Let $d$ be the cyclotomic derivation of $k[X]$, and let $\Delta$ be the factorisable derivation of $k[Y]$ associated with $d$, that is, $d(x_j)=x_{j+1}$ and $\Delta(y_j)=y_j(y_{j+1}-y_j)$ for all $j\in\mathbb Z_n$. We describe polynomial constants and rational constants of these derivations. We prove, among others, that the field of constants of $d$ is a field of rational functions over $k$ in $n-\f(n)$ variables, and that the ring of constants of $d$ is a polynomial ring if and only if $n$ is a power of a prime. Moreover, we show that the ring of constants of $\Delta$ is always equal to $k[v]$, where $v$ is the product $y_0... y_{n-1}$, and we describe the field of constants of $\Delta$ in two cases: when $n$ is power of a prime, and when $n=p q$.