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

Abstract

Let K be a differential field of zero characteristic with a basic set of derivations Δ = {δ 1, …, δ m } and let Θ denote the free commutative semigroup of all elements of the form \( \theta = \delta_1^{{k_1}} \cdots \delta_m^{{k_m}} \) where k i ∈ ℕ (1 ≤ i ≤ m). Let the order of such an element be defined as ord \( \theta = \sum\limits_{i = 1}^m {{k_i}} \), and for any r ∈ ℕ, let Θ(r) = {θ ∈ Θ | ord θ ≤ r}. Let L = K〈η 1, …, η s 〉 be a differential field extension of K generated by a finite set η = {η 1, …, η s } and let F be an intermediate differential field of the extension L/K. Furthermore, for any r ∈ ℕ, let \( {L_r} = K\left( {\bigcup\limits_{i = 1}^s {\Theta (r){\eta_i}} } \right) \) and F r = L r ∩ F. We prove the existence and describe some properties of a polynomial ϕ K,F,η (t) ∈ ℚ[t] such that ϕ K,F,η (r) = trdeg K F r 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.