Dimension polynomials of intermediate differential fields and the strength of a~system of differential equations with group action

A. B. Levin

Fundamentalʹnaâ i prikladnaâ matematika, Tome 14 (2008) no. 4, pp. 167-180

Résumé

Let $K$ be a differential field of zero characteristic with a basic set of derivations $\Delta=\{\delta_1,\dots,\delta_m\}$ and let $\Theta$ denote the free commutative semigroup of all elements of the form $\theta=\delta_1^{k_1}\dots\delta_m^{k_m}$ where $k_i\in\mathbb N$ ($1\leq i\leq m$). Let the order of such an element be defined as $\operatorname{ord}\theta=\sum_{i=1}^mk_i$, and for any $r\in\mathbb N$, let $\Theta(r) = \{\theta\in\Theta\mid\operatorname{ord}\theta\leq r\}$. Let $L=K\langle\eta_1,\dots,\eta_s\rangle$ be a differential field extension of $K$ generated by a finite set $\eta=\{\eta_1,\dots,\eta_s\}$ and let $F$ be an intermediate differential field of the extension $L/K$. Furthermore, for any $r\in\mathbb N$, let $L_r=K\Bigl(\bigcup_{i=1}^s\Theta(r)\eta_i\Bigr)$ and $F_r=L_r\cap F$. We prove the existence and describe some properties of a polynomial $\varphi_{K,F,\eta}(t)\in\mathbb Q[t]$ such that $\varphi_{K,F,\eta}(r)=\operatorname{trdeg}_KF_r$ for all sufficiently large $r\in\mathbb N$. This result implies the existence of a dimension polynomial that describes the strength of a system of differential equations with group action in the sense of A. Einstein. We shall also present a more general result, a theorem on a multivariate dimension polynomial associated with an intermediate differential field $F$ and partitions of the basic set $\Delta$.