FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2008, VOLUME 14, NUMBER 4, PAGES 167-180

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

A. B. Levin

Abstract

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Let $K$ be a differential field of zero characteristic with a basic set of derivations $$D = {d₁,...,d_m} and let $$Q denote the free commutative semigroup of all elements of the form $$q = d₁^k₁...d_m^k_m where $k$_i Î N ($1$£ i £ m). Let the order of such an element be defined as $ord$q = å_i=1^m k_i, and for any $r$Î N, let $$Q(r) = {q Î Q | ord q £r}. Let $L\; =\; K$áh₁,...,h_sñ be a differential field extension of $K$ generated by a finite set $$h = {h₁,...,h_s} and let $F$ be an intermediate differential field of the extension $L/K$. Furthermore, for any $r$Î N, let $L$_r = K(È_i=1^s Q(r) h_i) and $F$_r = L_r Ç F. We prove the existence and describe some properties of a polynomial $$j_K,F,h(t) Î Q[t] such that $$j_K,F,h(r) = trdeg_KF_r for all sufficiently large $r$Î N. This result implies the existence of a dimension polynomial that describes the strength of a system of differential equations with group action in the sense of A. Einstein. We shall also present a more general result, a theorem on a multivariate dimension polynomial associated with an intermediate differential field $F$ and partitions of the basic set $$D.

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