Counting solutions of differential equations
- Zählen von Lösungen von Differentialgleichungen
Lange-Hegermann, Markus; Plesken, Wilhelm (Thesis advisor)
Dissertation / PhD Thesis
Aachen, Techn. Hochsch., Diss., 2014
Systems of differential equations are notoriously hard to solve, and many such systems do not admit closed form solutions in "elementary" functions. Despite this, increasingly good heuristics are implemented in computer algebra systems to find solutions. Given a set of closed form solutions returned by a computer algebra system, the question remains whether this set is the complete solution set. The aim of this thesis is a quantitative analysis of the solution set of a system of differential equations, which decides whether the solutions found by a heuristical solver form a proper subset of the complete solution set. Therefore, this thesis examines three measures of the size of the solution set of a system of differential equations: the differential dimension polynomial, the counting sequence, and the differential counting polynomial. The differential dimension polynomial was originally introduced by Kolchin to describe the size of solution sets of a prime differential ideal in the sense that it generically describes the number of free power series coefficients up to any order. This thesis generalizes the differential dimension polynomial and its invariance conditions under differential birational maps from differential prime ideals to ideals associated to so-called simple differential systems. The differential dimension polynomial carries enough information to reliably answer the question whether two full solution sets of ideals associated to simple differential systems included in each other are equal, and this sufficiently describes most common differential systems. To give a non-generic description of the size of the solution set of a system of differential equations, this thesis introduces the counting sequence. The counting sequence describes the set of Taylor polynomials of solutions of each degree precisely, in the sense that it accounts for finite and countably infinite exceptional sets. If there exists a closed form polynomial that ultimately describes the counting sequence, then this closed form is called the differential counting polynomial. It is well-known that there cannot be an algorithm to compute the counting sequence or the differential counting polynomial. Nevertheless, in this thesis the counting sequence and the differential counting polynomial are computed for many important classes of systems of differential equations, in particular linear systems, most common semilinear systems, and quasilinear first order ordinary differential equations; in particular, for these classes of differential equations the existence of both the counting sequence and the differential counting polynomial is proved. Both these measures can decide whether an inclusion of two solution sets is proper under the condition that no countably infinite exceptional sets appear. The differential dimension polynomial, the counting sequence, and the differential counting polynomial determine classical measures that describe the size of the solution set of a system of differential equations, including the number of free functions, Cartan's characters and index of generality, Einstein's strength, and classical invariants from differential algebra like the differential type, the differential dimension, and the typical dimension. The Thomas decomposition algorithm, which is implemented as part of this thesis, is the algorithmic foundation for these descriptions of the size of solution sets. This algorithm partitions the solution set into solution sets of simple differential systems. It allows to compute the differential dimension polynomial and also certain consequences of differential systems, which are independent of counting.