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, $$C< x, \partial >^H=\{f^{\partial}< x> \in C< x, \partial> : f^{\partial}< hx+ h_0> = f^{\partial}< x> {whenever} (h,h_0)\in H \},$$ $$C< x, \partial>^{(GL^{\partial}(m,F),H)}=\{f^{\partial}< x> \in C< x, \partial> : f^{g^{-1}\partial}< hx+ h_0> = f^{\partial}< x> {whenever} g\in GL^{\partial}(m,F) {and} (h,h_0)\in H \}$$ 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 $$GL^{\partial}(m,F)= \{g=(g_{jk})_{j,k=\bar{1,m}}\in GL(m,F): \partial_ig_{jk}= \partial_jg_{ik} {for} i,j,k=\bar{1,m}\}$$, $\partial$ stands for the column-vector with the "coordinates" $\partial_1,. . >.,\partial_m$. In the paper these two fields are described.