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,...,dm} and let Q denote the free commutative semigroup of all elements of the form q = d1k1...dmkm where ki Î N (1 £ i £ m). Let the order of such an element be defined as ord q = åi=1m ki, and for any r Î N, let Q(r) = {q Î Q | ord q £r}. Let L = Káh1,...,hsñ be a differential field extension of K generated by a finite set h = {h1,...,hs} and let F be an intermediate differential field of the extension L/K. Furthermore, for any r Î N, let Lr = K(Èi=1s Q(r) hi) and Fr = Lr Ç F. We prove the existence and describe some properties of a polynomial jK,F,h(t) Î Q[t] such that jK,F,h(r) = trdegKFr 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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