Chapter 1. Introduction

Abstract

In this book we will be mainly concerned with the analysis of “general” systems of differential equations. As there are many ways how one might interpret this “generality”, we should start by clarifying what we mean by this term. The first differential equation any student of mathematics (or of some natural or engineering sciences) encounters is most probably a scalar ordinary differential equation of the form \(u^\prime = \phi(x,u)\) where \(u=u(x)\) is the unknown function to be determined.

Entering the theory of such equations, one rapidly notices that the restriction to scalar equations is unnecessary at most places. Almost all proofs remain valid, if we generalise to systems of the form \({\bf u^\prime} = {\bf \phi}(x,{\bf u})\) where now \({\bf u} = (u^1,\ldots,u^m)\) is a vector of unknown functions.¹ A crucial implication of the assumed form is that our system contains as many equations as unknowns. Under rather modest assumptions on the function \({\bf \phi}\) (e. g. Lipschitz continuity), the classical existence and uniqueness theorems assert that such systems are.at least locally.uniquely solvable for arbitrary choices of the initial data \({\bf u}(0) = {\bf u}_0\) with a constant vector \({\bf u}_0 \in \mathbb{R}^m.\). Extending a terminology usually only applied to partial differential equations, we call systems of the above form normal.

While the majority of analytical or numerical approaches to differential equations is exclusively concerned with normal systems, many applications lead to a more general form, namely implicit systems \({\bf \Phi}(x, {\bf u,u^\prime}) = 0.\). If we assume that the Jacobian \(\partial\Phi/\partial{\bf u^\prime}\) is everywhere regular, then the Implicit Function Theorem tells us that any such system is at least locally equivalent to a normal one (whether one can effectively transform it into solved form is a very different question, but theoretically it is always possible). Obviously, a necessary condition for regularity is that the Jacobian is a square matrix, i. e. that there are as many equations as unknown functions contained in the system. After the transformation we can again apply the familiar existence and uniqueness theorems.

Werner M. Seiler

Chapter 2. Formal Geometry of Differential Equations

Abstract

In this chapter we lay the geometric foundations of the formal theory of differential equations. The name “formal theory” stems from the fact that it is, at least indirectly, concerned with the analysis of formal power series solutions, i. e. one ignores the question whether the series actually converge. Another interpretation of the name is that one tries to extract as much information as possible on the solution space by purely formal operations like algebraic manipulations of the equations or their differentiations without actually solving the given equation.

The basic tool for a geometric approach to differential equations is the jet bundle formalism and the first two sections give an introduction to it. We do this first in a more pedestrian way considering jets as a differential geometric approach to (truncated) power series. For most computational purposes this simple point of view is sufficient. In order to obtain a deeper understanding of certain structural properties which will later be of importance, we redevelop the theory in the second section in a more abstract but intrinsic way which does not require power series. In both approaches, special emphasis is put on the contact structure as the key to the geometry of jet bundles. Because of its great importance, we consider different geometric realisations of it, each having its advantages in certain applications.

Werner M. Seiler

Chapter 3. Involution I: Algebraic Theory

Abstract

The first chapter was mainly concerned with the geometry behind the formal theory. For many problems a purely geometric approach is not sufficient (a concrete example is the question of proving the formal integrability of a differential equation) and additional algebraic ingredients are needed. These lead us to the title concept of this book: involution. Its algebraic theory will be the topic of the next four chapters. We will start in the current chapter with a more combinatorial approach leading to a special kind of Gröbner bases, the involutive bases; its algorithmic realisation is the topic of Chapter 4. In Chapter 5 we will show that the structure analysis of polynomial modules becomes significantly easier with the help of such bases. Finally, Chapter 6 will provide us with a homological explanation of the remarkable properties of (some) involutive bases. In Chapter 7 we will then return to differential equations and see how these algebraic ingredients appear there naturally in the analysis of the symbol of a differential equation.

In the first section we introduce the notion of an involutive division, a restriction of the usual divisibility relation of power products. As we are interested in applications beyond the classical commutative case, i. e. also in situations where the variables do not commute with each others, we introduce the theory for the monoid of multi indices and not for terms. Here we will already meet most of the key ideas about involution, as the later extension to polynomials will be fairly straightforward.

Werner M. Seiler

Chapter 4. Completion to Involution

Abstract

In the previous chapter we only defined the notion of an involutive basis but we did not consider the question of the existence of such bases. Recall from Section 3.3 that (in the case of a coefficient field) the existence proof for Gröbner bases is straightforward. For involutive bases the situation is considerably more complicated. Indeed, we have already seen in Example 3.1.16 an (even monomial) ideal not possessing a finite Pommaret basis. Thus we surely cannot expect that an arbitrary polynomialideal has for every involutive division a finite involutive basis.

In Section 4.1 we introduce a special class of involutive divisions, the constructive divisions, which is naturally related to an algorithm for computing involutive bases (contained in Section 4.2). If such a division is in addition Noetherian, then the algorithm will always terminate with an involutive basis and thus provides us with a proof of the existence of such bases for many divisions (including in particular the Janet division). Unfortunately, both the definition of constructive divisions and the termination proof are highly technical and not very intuitive.

As a reward the underlying algorithm turns out to be surprisingly efficient despite its simplicity. However, in general, it does not produce a minimal basis and furthermore still contains some redundancies. In Section 4.4 we show how it can be modified such that the output is always minimal and simultaneously introduce a number of optimisations. This optimised algorithm underlies most implementations of involutive bases in computer algebra systems.

Werner M. Seiler

Chapter 5. Structure Analysis of Polynomial Modules

Abstract

We now apply the theory of involutive bases developed in Chapter 3 to the structure analysis of modules over the commutative polynomial ring, i. e. we do some classical commutative algebra (the question to what extent the results presented here also hold for more general polynomial rings of solvable type is still open and will not be discussed here). The basic observation is that the Pommaret basis with respect to the degree lexicographic term order provides us with an easy access to several interesting invariants; more precisely, this basis is to a considerable extent determined by the structure of the module. As this particular type of basis will also play an important role in the analysis of differential equations, this fact simultaneously allows us an algebraic interpretation of many aspects of the theory of differential equations.

We start in the first section by analysing combinatorial or Stanley decompositions: a polynomial module is written as a direct sum of free modules over polynomial rings in a restricted set of variables. It is a natural consequence of our definition of involutive bases (and in fact the main motivation for it) that any involutive basis of a submodule of a free polynomial module immediately induces such a decomposition. For more general modules the situation is more complicated. Assuming that we deal with a finitely generated module, we can present it as the quotient of a free polynomial module by a submodule and then construct a complementary decomposition to a Gröbner basis of the submodule.

Werner M. Seiler

Chapter 6. Involution II: Homological Theory

Abstract

A reader with some experience in commutative algebra has probably noticed that almost all the invariants of polynomial modules which we computed in the last chapter with the help of Pommaret bases are actually of a homological origin. Hence one suspects that this special type of involutive basesmust be closely related to some homological constructions. The main purpose of this chapter is to demonstrate that such a relationship indeed exists.

The first section introduces the two relevant complexes: the polynomial de Rham and the Koszul complex, respectively, over a finite-dimensional vector space. In particular, we study the duality between them and prove that they are exact in positive degree (and hence define a (co)resolution). Then we restrict these complexes to certain subcomplexes. In the case of the Koszul complex, this restriction is simply achieved by tensoring it with some polynomial module leading to the familiar Koszul homology of the module.

For the polynomial de Rham complex one introduces classically the notion of a symbolic system. As we will see, such a system is nothing but a polynomial comodule, if one exploits the fact that the de Rham complex can be equally well defined over the symmetric coalgebra. Taking this point of view, the Spencer cohomology of a symbolic system is obtained completely dually to the Koszul homology by cotensoring the polynomial de Rham complex with a comodule. Finally, we define a (co)module to be involutive at a certain degree, if the corresponding bigraded (co)homology vanishes beyond this degree.

Werner M. Seiler

Chapter 7. Involution III: Differential Theory

Abstract

In this chapter we return again to the study of differential equations. We will now combine the geometric theory introduced in Chapter 2 with the algebraic and homological constructions of the last four chapters in order to arrive finally at the notion of an involutive equation. The key is the (geometric) symbol \(\mathcal{N}_q \subseteq V\pi^q_{q-1}\) of an equation \( \mathcal{R}_q \subseteq J_q \pi \) which we define in the first section. The fundamental identification \(V\pi_{q-1}^q \cong S_q(T^\ast \mathcal{X}) \mathop\otimes\limits_{J_{q-1}\pi} V \pi\) discussed in Section 2.2 builds a bridge between the geometric and the algebraic side of formal theory.

Intrinsically, the symbol defines via prolongation at each point on \( \mathcal{R}_q \) a subcomodule of a free comodule over the symmetric coalgebra, i. e. a symbolic system, to which we may apply the homological theory of Chapter 6. Dually, the symbol equations generate a submodule of a free module over the symmetric algebra. In local coordinates we may identify the symmetric algebra with the polynomial algebra and thus are exactly in the situation treated in Chapter 5.

Either way, the algebraic theory leads naturally to the notion of an involutive symbol. For readers not so familiar with abstract algebra, we will repeat some results from the previous chapters in differential equations terminology. In particular, we will discuss a concrete criterion for an involutive symbol which is useful in coordinate computations. However, it is valid only in δ-regular coordinates so that it must be applied with some care.

Werner M. Seiler

Chapter 8. The Size of the Formal Solution Space

Abstract

The results in this chapter should be used with some care.We introducemeasures for the size of the formal solution space of a differential equation. First of all, one cannot stress strongly enough that these considerations only concerns formal solutions, i. e. formal power series satisfying the given equation. In contrast to Chapter 9, we do not bother here about the convergence of these series and thus the results do not imply any statement on the existence of strong solutions. So one might say that we perform in this chapter only combinatorial games. Nevertheless, there are situations where the results can be quite useful. This should be evident from the fact that scientists like Cartan and Einstein actively participated in these “games”.

In the first section we briefly discuss the concept of a “general” solution. This word is often used but hardly anybody cares to give a rigorous definition.We do not give one either, as this turns out to be rather difficult and of doubtful value. So we only discuss some of the arising problems in order to motivate why we afterwards restrict to the formal solution space where the situation is much simpler.

Section 8.2 introduces the main tools of the formal theory for measuring the size of the formal solution space: the Cartan characters and the Hilbert function. We exhibit the connection between them and show how they can be explicitly determined for an involutive equation. For a classical analyst these concepts may appear to be rather abstract measures. Therefore we also discuss how—in certain circumstances—they can be translated into statements about how many functions are needed for parametrising the solution space.

Werner M. Seiler

Chapter 9. Existence and Uniqueness of Solutions

Abstract

A fundamental issue in the theory of differential equations consists of proving the existence and uniqueness of solutions. Before we discuss partial differential equations, we analyse in Section 9.1 the situation for general systems of ordinary differential equations, often called differential algebraic equations. For involutive equations it is here straightforward to extend the classical existence and uniqueness theorem. The formal theory also provides us with a natural geometric approach to the treatment of certain types of singularities.

Traditionally, (first-order) ordinary differential equations are studied via vector fields on the manifold \(\mathcal{E}\) (actually, one usually restricts to the autonomous case assuming that \(\mathcal{E} = \mathcal{X} \times \mathcal{U}\) and considers vector fields on \(\mathcal{U}\)). However, for a unified treatment of many singular phenomena it turns out to be much more useful to associate with the equation a vector field (or more precisely a distribution) in the first jet bundle \(J_1 \pi\) arising very naturally from the contact structure. We will not develop a general theory of singularities but study a number of situations that have attracted much interest in the literature.

In local coordinates, one may say that the study of power series solutions underlies much of the formal theory. Hence, it is not surprising that results on analytic solutions of partial differential equations are fairly straightforward to obtain. In Section 9.2 we recall the famous Cauchy—Kovalevskaya Theorem for normal systems. The main point of the proof consists of showing that the easily obtained formal power series solution of the usual initial value problem actually converges.

Werner M. Seiler

Chapter 10. Linear Differential Equations

Abstract

Linear differential equations are simpler in many respects. The truth of this statement is already obvious from the fact that their solution spaces possess the structure of a vector space. Thus it is not surprising that some of our previous results may be improved in this special case. In the first section we study how the linearity can be expressed within our geometric framework for differential equations. This topic includes in particular a geometric formulation of the linearisation of an arbitrary equation along one of its solutions.

In the discussion of the Cartan–Kähler Theorem in Section 9.4 we emphasised that the uniqueness statement holds only within the category of analytic functions; it is possible that further solutions with lower regularity exist. In the case of linear systems stronger statements hold. In Section 10.2 we will use our proof of the Cartan–Kähler Theorem to extend the classical Holmgren Theorem on the uniqueness of \(\mathcal{C}^1\) solutions from normal equations to arbitrary involutive ones.

A fundamental topic in the theory of partial differential equations is the classification into elliptic and hyperbolic equations. We will study the notion of ellipticity in Section 10.3 for arbitrary involutive equations. One of our main results will be that the approach to ellipticity via weights usually found in the literature is not only insufficient but also unnecessary, if one restricts to involutive equations: if a system is elliptic with weights, then its involutive completion is also elliptic without weights; the converse is not true.

Werner M. Seiler

Appendix A. Miscellaneous

Abstract

The first section of this chapter fixes some notations for multi indices that are widely used in the main text. Except for the distinction into multi and “repeated” indices these are standard notations in multivariate analysis. We also show an important finiteness result for multi indices, namely Dickson’s Lemma, and discuss term orders. The next section recalls those properties of (real-)analytic functions that are needed in the proof of the Cauchy–Kovalevskaya Theorem. Then some elementary operations with differential equations like reduction to first order or quasilinearisation are discussed within the context of the formal theory. Finally, a few simple facts about the modified Stirling numbers appearing in manipulations of Cartan characters and Hilbert functions are presented.

Werner M. Seiler

Appendix B. Algebra

Abstract

A proper understanding of the meaning of the concept of involution in the form introduced in Chapter 3 requires certain ideas from (commutative) algebra collected in this chapter. Some classical references for all mentioned topics are [98, 99, 125]; older textbooks on commutative algebra are usually less constructive. The rather new books [185, 267, 268] put strong emphasis on computational issues (using the computer algebra systems Singular and CoCoa, respectively). Basic algebraic notions are introduced in [281, 415]; for the theory of non-commutative rings we mention [171, 276, 319].

The first section quickly reviews some basic algebraic structures like monoids, rings, algebras etc. Much of the material is elementary. Special emphasis is put on modules, as they are central for the algebraic analysis of the symbol of a differential equation. In particular, we introduce the Hilbert function and polynomial, as similar concepts appear for differential equations in Section 8.2. A simple method for the determination of the Hilbert function may be considered as one motivation for the introduction of involutive bases.

The second section reviews a few basic concepts from homological algebra. We introduce (co)homology modules for complexes of \( \mathcal{R} \)-modules and discuss resolutions. These notions are in particular used in Chapters 5 and 6. Section 10.5 requires in addition some knowledge about exact and derived functors. The third section concerns the more specialised topic of coalgebras and comodules; these concepts are only used in Chapter 6.

Werner M. Seiler

Appendix C. Differential Geometry

Abstract

As in this book we are much concerned with a geometric approach to differential equations, we need some basic notions in differential geometry which are collected in this chapter. We start with a section introducing (differentiable) manifolds and some spaces associated with them: the tangent and the cotangent bundle. Special emphasis is given to fibred manifolds (bundles are only briefly introduced, as we hardly need their additional structures). The second section studies in some more detail vector fields and differential forms and the basic operations with them. Distributions of vector fields (or dual codistributions of one-forms, respectively) and the Frobenius theorem are the topic of the third section.

A fundamental object in differential geometry are connections. In our context, it should be noted that ordinary differential equations and certain partial differential equations correspond geometrically to connections (see Remark 2.3.6). The fourth section introduces connections on arbitrary fibred manifolds. At a few places we need some elementary results about Lie groups and algebras which are collected in the fifth section. The final section is concerned with (co)symplectic geometry as the basis of the theory of Hamiltonian systems.

Standard references on differential geometry are [3, 49, 81, 256, 282, 474], but they go much further than we need. For most of our purposes a working knowledge of some basic notions around manifolds is sufficient. Much of the material we need can also be found in a very accessible form in [342, Chapt. 1].

Werner M. Seiler

Springer Professional

Involution

The Formal Theory of Differential Equations and its Applications in Computer Algebra

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Chapter 2. Formal Geometry of Differential Equations

Chapter 3. Involution I: Algebraic Theory

Chapter 4. Completion to Involution

Chapter 5. Structure Analysis of Polynomial Modules

Chapter 6. Involution II: Homological Theory

Chapter 7. Involution III: Differential Theory

Chapter 8. The Size of the Formal Solution Space

Chapter 9. Existence and Uniqueness of Solutions

Chapter 10. Linear Differential Equations

Appendix A. Miscellaneous

Appendix B. Algebra

Appendix C. Differential Geometry

Backmatter