A note on the kernels of higher derivations

Li, Jiantao ; Du, Xiankun

Czechoslovak Mathematical Journal, Tome 63 (2013) no. 3, pp. 583-588

Résumé

Let $k\subseteq k'$ be a field extension. We give relations between the kernels of higher derivations on $k[X]$ and $k'[X]$, where $k[X]:=k[x_1,\dots ,x_n]$ denotes the polynomial ring in $n$ variables over the field $k$. More precisely, let $D=\{D_n\}_{n=0}^\infty $ a higher $k$-derivation on $k[X]$ and $D'=\{D_n'\}_{n=0}^\infty $ a higher $k'$-derivation on $k'[X]$ such that $D'_m(x_i)=D_m(x_i)$ for all $m\geq 0$ and $i=1,2,\dots ,n$. Then (1) $k[X]^D=k$ if and only if $k'[X]^{D'}=k'$; (2) $k[X]^D$ is a finitely generated $k$-algebra if and only if $k'[X]^{D'}$ is a finitely generated $k'$-algebra. Furthermore, we also show that the kernel $k[X]^D$ of a higher derivation $D$ of $k[X]$ can be generated by a set of closed polynomials.

Détail

DOI : 10.1007/s10587-013-0041-1

Classification : 13A50
Keywords: higher derivation; field extension; closed polynomial