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

Let K be a differential field of zero characteristic with a basic set of derivations Δ = {δ1, ..., δm} and let Θ denote the free commutative semigroup of all elements of the form where. Let the order of such an element be defined as ord, and for any r ∈ ℕ, let Θ(r) = {θ ∈ Θ {pipe} ord θ ≤ r}. Let L = K〈η1, ..., ηs〉 be a differential field extension of K generated by a finite set η = {η1, ..., ηs} and let F be an intermediate differential field of the extension L/K. Furthermore, for any r ∈ ℕ, let and Fr = Lr ∩ F. We prove the existence and describe some properties of a polynomial φ{symbol}K,F,η(t) ∈ ℚ[t] such that φ{symbol}K,F,η(r) = trdegKFr for all sufficiently large r ∈ 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 Δ. © 2009 Springer Science+Business Media, Inc.