Effective bounds for the consistency of differential equations

Abstract

One method to determine whether or not a system of partial differential equations is consistent is to attempt to construct a solution using merely the "algebraic data" associated to the system. In technical terms, this translates to the problem of determining the existence of regular realizations of differential kernels via their possible prolongations. In this paper we effectively compute an improved upper bound for the number of prolongations needed to guarantee the existence of such realizations, which ultimately produces solutions to many types of systems of partial differential equations. This bound has several applications, including an improved upper bound for the order of characteristic sets of prime differential ideals. We obtain our upper bound by proving a new result on the growth of the Hilbert-Samuel function, which may be of independent interest.

Estimates for the coefficients of differential dimension polynomials

“…The proof of the theorem makes use of the theory of differential kernels from [5]. Let us review the results that we will need.…”

“…Moreover, if M is compressed and ord ξ ≤ d for all ξ ∈ M , then Macaulay's theorem gives a sufficient condition for the Hilbert-Samuel function to have maximal growth, we will also make use of the following theorem that gives a necessary condition to have maximal growth (which appears as Corollary 4.9 in [5]). For ξ = (u 1 , .…”

“…The above proposition, together with Fact 2.1, essentially shows that, in order to produce the desired bound for the coefficients of the Kolchin polynomial, all that is missing is an upper bound for the order of a characteristic set of a prime differential ideal that depends solely on the order of the differential system. Such a bound was obtained in [5], its recursive formula uses the Ackermann function. We recall that this (nonprimitive recursive) function is defined as A :…”

Effective definability of Kolchin polynomials

“…It is easy to show, by induction say, that if one writes (t+1) · · · (t+m) = m j=0 c j t j , then c j ≤ 2 m−1 m!. From this it easy to show, by induction and using (5), that if we write ω Wi as m j=0 b i,j x j , then |b i,j | ≤ n2 m m!D m . Also note that, because the a i,j 's are integers, all the products b i,j · m!…”

Proof mining and effective bounds in differential polynomial rings

“…Recently there has been considerable interest in analyzing the effective content of the differential Nullstellensatz [9,20,22,24,38], with the methods employed coming from algebra and model theory [13,41,46]. Other constructive problems in differential algebra and differential algebraic geometry have also gained attention [5,14,15,21,23,28,39].…”