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

Abstract

An ordinary differential field $(F,d)$ 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, $$C<x,d{>}^H=\{f^d<x>\in C<x,d>:f^d<hx+h_0>=f^d<x>forany(h,h_0)\in H\},$$ $$C<x,d{>}^{(F^{\ast },H)}=\{f^d<x>\in C<x,d>:f^{g^{-1}d}<hx+h_0>=f^d<x>foranyg\in F^{\ast }and(h,h_0)\in H\}$$ are considered, where $C$ is the constant field of $(F,d)$ and $C<x,d>$ is the field of differential rational functions in $x_1,x_2,...,x_n$ over $C$. The field $C<x,d{>}^H$ ($C<x,d{>}^{(F^{\ast },H)}$) is an important tool in the equivalence problem of paths(respect. curves) in Differential Geometry with respect to the motion group $H$. In this paper an pure algebraic approach is offered to describe these fields. The field $C<x,d{>}^{(F^{\ast },H)}$ and its relation with $C<x,d{>}^H$ are investigated. It is shown also that $C<x,d{>}^H$ can be derived from some algebraic (without derivatives) invariants of $H$. Key words: Differential field, differential rational function, invariant, differential transcendent degree. 2000 Mathematics Subject Classification: 12H05, 53A04, 53A55