On the field of differential rational invariants of a subgroup of affine group (Partial differential case)

Abstract

An differential field $(F;{\partial }_1,...,{\partial }_m)$ of characteristic zero, a subgroup $H$ of affine group $GL(n,C)\propto C^n$ with respect to its identical representation in $F^n$ and the following two fields of differential rational functions in $x=(x_1,x_2,...,x_n)$-column vector, $$C<x,\partial {>}^H=\{f^{\partial }<x>\in C<x,\partial >:f^{\partial }<hx+h_0>=f^{\partial }<x>whenever(h,h_0)\in H\},$$ $$C<x,\partial {>}^{(G{L}^{\partial }(m,F),H)}=\{f^{\partial }<x>\in C<x,\partial >:f^{g^{-1}\partial }<hx+h_0>=f^{\partial }<x>wheneverg\in G{L}^{\partial }(m,F)and(h,h_0)\in H\}$$ are considered, where $C$ is the constant field of $(F,\partial )$, $C<x,\partial >$ is the field of $\partial$-differential rational functions in $x_1,x_2,...,x_n$ over $C$ and $$G{L}^{\partial }(m,F)=\{g=(g_{jk}{)}_{j,k=\overline{1,m}}\in GL(m,F):{\partial }_i{g}_{jk}={\partial }_j{g}_{ik}fori,j,k=\overline{1,m}\}$$, $\partial$ stands for the column-vector with the "coordinates" ${\partial }_1,..>.,{\partial }_m$. In the paper these two fields are described.