Involution. The formal theory of differential equations and its applications in computer algebra.

*(English)*Zbl 1205.35003
Algorithms and Computation in Mathematics 24. Berlin: Springer (ISBN 978-3-642-01286-0/hbk; 978-3-642-01287-7/ebook). xxii, 650 p. (2010).

The book is a comprehensive monograph describing the formal theory of differential equations in theoretical and computational aspects. It is devoted to an exposition of classical and modern results on formal methods in the investigation of general differential equations, including ordinary and partial, overdetermined and underdetermined equations, non-normal systems, implicit differential-algebraic equations, etc. The author stresses the constructive nature of the methods and puts a special emphasis on algorithms that can be effectively implemented into symbolic software.

The notion of involution in the book unites the algebraic, geometric and combinatorial aspects of the formal theory of differential equations. A significant part of the book is devoted to Pommaret bases, which are special kinds of Gröbner bases with specification for differential systems; in particular, they are useful when working with Koszul homology and Spencer cohomology. Cartan-Kähler theory, which is the general method to formulate a Cauchy problem, is presented in the book in its dual Vessiot version. These two features have mostly been presented in article form before, and so are described in this book in full detail for the first time.

Let us present the contents of the book.

The first chapter is an introduction that presents a range of sources for various types of differential equations and gives the motivation for a formal investigation of such equations. Different points of view of the mathematical and the physical community are underlined.

Chapter 2 concerns the geometry of jet spaces and differential equations. The author chooses two approaches to this classical subject – computational via coordinates and power series, and theoretical, where such important notions as affine structure in jet-fibers and prolongations are discussed.

The third chapter elaborates upon the algebraic theory, which extends the classical Janet-Riquier theory. It introduces involutive division on a monoid (with partial cases being Thomas, Janet and Pommaret divisions) with the emphasis on applications to non-commutative rings. These rings are studied later, with the focus on polynomial algebras of solvable type, that include the algebra of linear differential operators, difference Ore algebras and deformation quantizations. The Hilbert basis theorem is proved in this context and then involutive bases are introduced, following Gerdt and Blinkov, as a special kind of Gröbner bases with additional combinatorial properties.

The algorithmic realization of this approach is the topic of Chapter 4. The main question here is the existence of involutive bases. At first, it is proved that constructive divisions with Noetherian property (like Janet division) possess finite involutive bases, and so the corresponding algorithm always terminates. This is enhanced in the direction of minimization and optimization afterwards. Pommaret division is not Noetherian, and indeed the completion algorithm does not necessarily terminate. It however terminates in a generic coordinate system, and the problem of determining whether a given coordinate system is good (\(\delta\)-regularity) is discussed.

In Chapter 5, the theory of involutive bases is applied to analyze the structure of polynomial modules over commutative polynomial rings, so it is mainly concerned with the classical commutative algebra. Stanley decompositions, Noether normalizations, resolutions and syzygy theory are considered. The division algorithms are related to the dimension theory; in particular it is shown that the degree of a Pommaret basis with respect to the degree reverse lexicographic term order is equal to the Castelnuovo-Mumford regularity of the module.

Chapter 6 is dedicated to a homological explanation of the combinatorial properties of the involutive bases. It starts with the polynomial de Rham complex, the formal Poincaré lemma, then studies symbolic modules and prolongations and then discusses the duality between Spencer cohomology and Koszul homology (attributed to Grothendieck and Mumford). Cartan’s test is discussed in both geometric and algebraic situations and then the notion of involution is introduced in the geometric case (exterior differential systems lie behind the symbolic picture), which is dual to the notion of Castelnuovo-Mumford regularity in the algebraic case (as established by Malgrange). In the homological language, the involution is equivalent to the vanishing of certain Spencer cohomology groups due to a result of Serre-Guillemin-Sternberg. The estimates for \(\delta\)-regular and quasi-regular bases are then related to the degree of a Pommaret basis.

In Chapter 7, the author applies the developed algebraic theory to differential equations, and analyzes the symbolic modules by the methods of commutative and homological algebra. He begins with the notion of the (principal) symbol of a system of PDEs and discusses its relation with commutative algebra. In particular, prolongations, characteristics and the notion of involution for differential equations are introduced. Then, the integrability (compatibility) conditions are discussed, which lead to the notion of formal integrability, and, as a very important particular case, involutive differential equations (differential systems) are studied. The author makes a clear distinction between formal integrability and involution. Then, with a number of particular cases and the general Cartan-Kuranishi theorem, the notion of completion is discussed from both the theoretical side and the side of algorithmic implementation.

Chapter 8 deals with the size of the formal solution space, this is what is informally called functional dimension, to measure the amount of general solutions. The basic numerology is expressed through Cartan characters in the exterior differential system approach and via Hilbert function in the homological approach with symbolic modules. The author separately discusses gauge symmetric equations and explains in detail Einstein’s approach to count the general solutions in comparison with Cartan’s method.

In Chapter 9 existence and uniqueness of local solutions to systems of differential equations are reviewed. If such a system is not reducible to ODEs the assumption of analyticity is imposed, or alternatively one can work with formal solutions represented via power series. The general Cartan-Kähler theorem, generalizing the classical Cauchy-Kovalevskaya theorem for normal systems, is discussed as well as well-posedness of the Cauchy initial value problem. The end of the chapter is concerned with the modern version of the Vessiot theory. It is dual to the Cartan-Kähler theory of exterior differential systems. This part is close to what is called formal-geometric theory of PDEs, as approached by Krasilshchik, Lychagin and Vinogradov (the author uses the term Vessiot distribution instead of Cartan distribution, etc).

Chapter 10 is devoted to linear differential equations, in which case many results have a finer form. At first Holmgren’s uniqueness theorem is generalized to linear involutive systems in Cartan-Kähler form. Then it is shown that Douglis-Nirenberg weighted ellipticity is equivalent to the usual ellipticity of the involutive completion of the system. Then, the author discusses hyperbolic equations and boundary value problems. Compatibility complexes are discussed on the basis of resolvent theory, with an application to the inverse syzygy problem. Finally linear PDEs with constant coefficients are considered for which (especially in the case of finite type) the solution space has a nice parametrization.

Three appendices more than 100 pages long with some basics from algebra, geometry, differential equations and combinatorics finish the exposition and they make this book into a self-contained account of the formal theory of differential equations, especially in its computational aspect.

The notion of involution in the book unites the algebraic, geometric and combinatorial aspects of the formal theory of differential equations. A significant part of the book is devoted to Pommaret bases, which are special kinds of Gröbner bases with specification for differential systems; in particular, they are useful when working with Koszul homology and Spencer cohomology. Cartan-Kähler theory, which is the general method to formulate a Cauchy problem, is presented in the book in its dual Vessiot version. These two features have mostly been presented in article form before, and so are described in this book in full detail for the first time.

Let us present the contents of the book.

The first chapter is an introduction that presents a range of sources for various types of differential equations and gives the motivation for a formal investigation of such equations. Different points of view of the mathematical and the physical community are underlined.

Chapter 2 concerns the geometry of jet spaces and differential equations. The author chooses two approaches to this classical subject – computational via coordinates and power series, and theoretical, where such important notions as affine structure in jet-fibers and prolongations are discussed.

The third chapter elaborates upon the algebraic theory, which extends the classical Janet-Riquier theory. It introduces involutive division on a monoid (with partial cases being Thomas, Janet and Pommaret divisions) with the emphasis on applications to non-commutative rings. These rings are studied later, with the focus on polynomial algebras of solvable type, that include the algebra of linear differential operators, difference Ore algebras and deformation quantizations. The Hilbert basis theorem is proved in this context and then involutive bases are introduced, following Gerdt and Blinkov, as a special kind of Gröbner bases with additional combinatorial properties.

The algorithmic realization of this approach is the topic of Chapter 4. The main question here is the existence of involutive bases. At first, it is proved that constructive divisions with Noetherian property (like Janet division) possess finite involutive bases, and so the corresponding algorithm always terminates. This is enhanced in the direction of minimization and optimization afterwards. Pommaret division is not Noetherian, and indeed the completion algorithm does not necessarily terminate. It however terminates in a generic coordinate system, and the problem of determining whether a given coordinate system is good (\(\delta\)-regularity) is discussed.

In Chapter 5, the theory of involutive bases is applied to analyze the structure of polynomial modules over commutative polynomial rings, so it is mainly concerned with the classical commutative algebra. Stanley decompositions, Noether normalizations, resolutions and syzygy theory are considered. The division algorithms are related to the dimension theory; in particular it is shown that the degree of a Pommaret basis with respect to the degree reverse lexicographic term order is equal to the Castelnuovo-Mumford regularity of the module.

Chapter 6 is dedicated to a homological explanation of the combinatorial properties of the involutive bases. It starts with the polynomial de Rham complex, the formal Poincaré lemma, then studies symbolic modules and prolongations and then discusses the duality between Spencer cohomology and Koszul homology (attributed to Grothendieck and Mumford). Cartan’s test is discussed in both geometric and algebraic situations and then the notion of involution is introduced in the geometric case (exterior differential systems lie behind the symbolic picture), which is dual to the notion of Castelnuovo-Mumford regularity in the algebraic case (as established by Malgrange). In the homological language, the involution is equivalent to the vanishing of certain Spencer cohomology groups due to a result of Serre-Guillemin-Sternberg. The estimates for \(\delta\)-regular and quasi-regular bases are then related to the degree of a Pommaret basis.

In Chapter 7, the author applies the developed algebraic theory to differential equations, and analyzes the symbolic modules by the methods of commutative and homological algebra. He begins with the notion of the (principal) symbol of a system of PDEs and discusses its relation with commutative algebra. In particular, prolongations, characteristics and the notion of involution for differential equations are introduced. Then, the integrability (compatibility) conditions are discussed, which lead to the notion of formal integrability, and, as a very important particular case, involutive differential equations (differential systems) are studied. The author makes a clear distinction between formal integrability and involution. Then, with a number of particular cases and the general Cartan-Kuranishi theorem, the notion of completion is discussed from both the theoretical side and the side of algorithmic implementation.

Chapter 8 deals with the size of the formal solution space, this is what is informally called functional dimension, to measure the amount of general solutions. The basic numerology is expressed through Cartan characters in the exterior differential system approach and via Hilbert function in the homological approach with symbolic modules. The author separately discusses gauge symmetric equations and explains in detail Einstein’s approach to count the general solutions in comparison with Cartan’s method.

In Chapter 9 existence and uniqueness of local solutions to systems of differential equations are reviewed. If such a system is not reducible to ODEs the assumption of analyticity is imposed, or alternatively one can work with formal solutions represented via power series. The general Cartan-Kähler theorem, generalizing the classical Cauchy-Kovalevskaya theorem for normal systems, is discussed as well as well-posedness of the Cauchy initial value problem. The end of the chapter is concerned with the modern version of the Vessiot theory. It is dual to the Cartan-Kähler theory of exterior differential systems. This part is close to what is called formal-geometric theory of PDEs, as approached by Krasilshchik, Lychagin and Vinogradov (the author uses the term Vessiot distribution instead of Cartan distribution, etc).

Chapter 10 is devoted to linear differential equations, in which case many results have a finer form. At first Holmgren’s uniqueness theorem is generalized to linear involutive systems in Cartan-Kähler form. Then it is shown that Douglis-Nirenberg weighted ellipticity is equivalent to the usual ellipticity of the involutive completion of the system. Then, the author discusses hyperbolic equations and boundary value problems. Compatibility complexes are discussed on the basis of resolvent theory, with an application to the inverse syzygy problem. Finally linear PDEs with constant coefficients are considered for which (especially in the case of finite type) the solution space has a nice parametrization.

Three appendices more than 100 pages long with some basics from algebra, geometry, differential equations and combinatorics finish the exposition and they make this book into a self-contained account of the formal theory of differential equations, especially in its computational aspect.

Reviewer: Boris S. Kruglikov (Tromsø)

##### MSC:

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

13-02 | Research exposition (monographs, survey articles) pertaining to commutative algebra |

13P10 | Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) |

13P20 | Computational homological algebra |

35-04 | Software, source code, etc. for problems pertaining to partial differential equations |

68W30 | Symbolic computation and algebraic computation |