A note on the kernels of higher derivations

@article{Li2013,
abstract = {Let $k\subseteq k^\{\prime \}$ be a field extension. We give relations between the kernels of higher derivations on $k[X]$ and $k^\{\prime \}[X]$, where $k[X]:=k[x_1,\dots ,x_n]$ denotes the polynomial ring in $n$ variables over the field $k$. More precisely, let $D=\lbrace D_n\rbrace _\{n=0\}^\infty $ a higher $k$-derivation on $k[X]$ and $D^\{\prime \}=\lbrace D_n^\{\prime \}\rbrace _\{n=0\}^\infty $ a higher $k^\{\prime \}$-derivation on $k^\{\prime \}[X]$ such that $D^\{\prime \}_m(x_i)=D_m(x_i)$ for all $m\ge 0$ and $i=1,2,\dots ,n$. Then (1) $k[X]^D=k$ if and only if $k^\{\prime \}[X]^\{D^\{\prime \}\}=k^\{\prime \}$; (2) $k[X]^D$ is a finitely generated $k$-algebra if and only if $k^\{\prime \}[X]^\{D^\{\prime \}\}$ is a finitely generated $k^\{\prime \}$-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.},
author = {Li, Jiantao, Du, Xiankun},
journal = {Czechoslovak Mathematical Journal},
keywords = {higher derivation; field extension; closed polynomial; higher derivation; field extension; closed polynomial},
language = {eng},
number = {3},
pages = {583-588},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A note on the kernels of higher derivations},
url = {http://eudml.org/doc/260598},
volume = {63},
year = {2013},
}

TY - JOUR
AU - Li, Jiantao
AU - Du, Xiankun
TI - A note on the kernels of higher derivations
JO - Czechoslovak Mathematical Journal
PY - 2013
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 63
IS - 3
SP - 583
EP - 588
AB - Let $k\subseteq k^{\prime }$ be a field extension. We give relations between the kernels of higher derivations on $k[X]$ and $k^{\prime }[X]$, where $k[X]:=k[x_1,\dots ,x_n]$ denotes the polynomial ring in $n$ variables over the field $k$. More precisely, let $D=\lbrace D_n\rbrace _{n=0}^\infty $ a higher $k$-derivation on $k[X]$ and $D^{\prime }=\lbrace D_n^{\prime }\rbrace _{n=0}^\infty $ a higher $k^{\prime }$-derivation on $k^{\prime }[X]$ such that $D^{\prime }_m(x_i)=D_m(x_i)$ for all $m\ge 0$ and $i=1,2,\dots ,n$. Then (1) $k[X]^D=k$ if and only if $k^{\prime }[X]^{D^{\prime }}=k^{\prime }$; (2) $k[X]^D$ is a finitely generated $k$-algebra if and only if $k^{\prime }[X]^{D^{\prime }}$ is a finitely generated $k^{\prime }$-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.
LA - eng
KW - higher derivation; field extension; closed polynomial; higher derivation; field extension; closed polynomial
UR - http://eudml.org/doc/260598
ER -