Estimates for the coefficients of differential dimension polynomials

Abstract

We answer the following long-standing question of Kolchin: given a system of algebraic-differential equations Σ(x 1 , . . . , xn) = 0 in m derivatives over a differential field of characteristic zero, is there a computable bound, that only depends on the order of the system (and on the fixed data m and n), for the typical differential dimension of any prime component of Σ? We give a positive answer in a strong form; that is, we compute a (lower and upper) bound for all the coefficients of the Kolchin polynomial of every such prime component. We then show that, if we look at those components of a specified differential type, we can compute a significantly better bound for the typical differential dimension. This latter improvement comes from new combinatorial results on characteristic sets, in combination with the classical theorems of Macaulay and Gotzmann on the growth of Hilbert-Samuel functions.

Algorithms yield upper bounds in differential algebra

“…Then [15,Proposition 3.1] and [15,Fact 2.1] imply that |ω Y | is bounded by a computable function of N .…”

“…there is a characteristic set of Y with respect to the orderly ranking with the order bounded by a computable function of N. Then, [15, Proposition 3.1] and[15, Fact 2.1] imply that |ω Y | is bounded by a computable function of N. ∎ Proposition 6 7. There exists an algorithm that, for every computable function g(n)∶ Z ⩾0 → Z ⩾0 , produces a number Len g such that, for every sequence of Kolchin polynomialsω 0 > ω 1 > ⋯ > ω such that |ω i | < g(i)for every 0 ⩽ i ⩽ , we have < Len g .Proof By replacing g(n) with n + max 0⩽s⩽n g(s), we can further assume that g(n) is increasing and g(n) ⩾ n. Definition 2.4.9 and Lemma 2.4.12 of[22] define a computable order-preserving map c from the set of all Kolchin polynomials K to Z m+1 ⩾0 (considered with respect to the lexicographic ordering)…”

Algorithms yield upper bounds in differential algebra

“…Since a characteristic set of Y can be obtained from C by selecting the polynomials only in the first ℓ variables, there is a charactersitic set of Y with respect to the orderly ranking with the order bounded by a computable function of N . Then [16, Proposition 3.1] and [16,Fact 2.1] imply that |ω Y | is bounded by a computable function of N .…”

Some properties of multivariate differential dimension polynomials and their invariants

“…These invariants are closely connected to some other important characteristics; for example, one of them is the differential transcendence degree of the extension. Among recent works on univariate differential dimension polynomials one has to mention the work of O. Sanchez [14] on the evaluation of the coefficients of a differential dimension polynomial, the work of J. Freitag, O. Sanchez and W. Li on the definability of Kolchin polynomials, and works of M. Lange-Hegermann [8] and [9], where the author introduced a differential dimension polynomial of a characterizable (not necessarily prime) differential ideal and a countable differential polynomial that generalizes the concept of differential dimension polynomial.…”