##
**Formal algorithmic elimination for PDEs.**
*(English)*
Zbl 1339.35007

Lecture Notes in Mathematics 2121. Cham: Springer (ISBN 978-3-319-11444-6/pbk; 978-3-319-11445-3/ebook). viii, 283 p. (2014).

This book evolved out of the habilitation thesis of the author and is concerned with formal computations with general systems of partial differential equations and inequations with polynomial nonlinearities. The term “formal” has here two meanings: it refers on one hand to the fact that one manipulates exclusively the equations themselves without any assumptions about function spaces in which solutions should live and it signals on the other hand that only formal (or – if convergence can be proven – analytic) solutions are considered. The approach taken in the book is based on differential algebra (more precisely, the differential ideal theory founded by Ritt): the central object is the ring of differential polynomials in finitely many differential unknowns and its (differential) ideals. Besides a short introductory chapter, the book consists of two chapters and an appendix. One chapter is concerned with the basic algorithmic tools used in the other chapter to solve certain differential elimination problems for analytic functions.

Two fundamental algorithms are used throughout the book: Janet bases and Thomas decompositions. Janet bases are a particular form of involutive bases which in turn are Gröbner bases with additional combinatorial properties which makes them very useful for many computations. They combine the classical Janet-Riquier theory of differential equations with the theory of Gröbner bases of polynomial ideals. Although Riquier and Janet developed originally their theory for nonlinear systems of differential equations, one obtains an algorithmic theory in the modern sense only for linear systems. Consequently, the author presents the theory of Janet bases in the context of Ore algebras. This class of non-commutative polynomial rings contains not only linear differential operators, but also linear difference or shift operators.

A Thomas decomposition deals with the problems posed by fully nonlinear systems where one usually must make case distictions depending on the initials and separants of the equations. As the case distinctions automatically lead to inequations, the decomposition is explained for systems composed of equations and inequations. First, the case of algebraic systems is treated; then the theory is extended to differential systems. Complete algorithms are exhibited and their implementation is discussed.

A number of different elimination problems are treated in the book. The first one is the classical algebraic problem of computing elimination ideals. It is considered here for ideals in an Ore algebra. In principle, this problem is solved by a Gröbner basis for the lexicographic term order. In practice, this approach is often not feasible within a reasonable amount of computing time. The author presents a technique which he calls degree steering and which consists of computing several Janet bases for different term orders. As another problem, the author also considers the question of “eliminating” standard basis vectors instead of variables for submodules of free modules. Also the classical task of finding the compatibility conditions for inhomogeneous systems of linear functional equations is treated here as an elimination problem.

The next type of problems has been term “linear differential elimination” by the author. Here one is given a family of complex analytic functions of the form \(\sum f_i(\alpha_i(z))g_i(z)\), where \(\alpha_i\) and \(g_i\) are prescribed and \(f_i\) arbitrary analytic functions. The task consists of deciding whether a further given analytic function \(u(z)\) is a member of this family and if yes to find corresponding parameters \(f_i\). The author solves this problem by obtaining an implicit representation of the family, namely a system of partial differential equations such that the family defines its solution space. Some applications to the construction of special solutions of system of differential equations are given.

Finally, a nonlinear version of this problem is treated where the given family of analytic functions is no longer a linear combination but of the general form \(p(f_1(\alpha_1(z),\dots,f_k(\alpha_k(z)))\). Again the basic idea is to obtain an implicit representation as solution space of a differential system. However, the nonlinearity makes it considerably harder to construct such a representation.

The second chapter on the used algorithmic tools mainly compiles and reviews results from the literature. But as in particular the Thomas decomposition is hardly known even to specialists in computer algebra, it is very useful to have this material presented in a uniform and succinct manner. The third chapter on elimination problems contains original results by the author. The book is well written with many pointers to the literature, in particular for those aspects that could not be covered in more details because of lack of space. All algorithms described in it have been implemented by the author and his collaborators in the computer algebra system Maple. For most of the packages, web addresses are given in the bibliography.

Two fundamental algorithms are used throughout the book: Janet bases and Thomas decompositions. Janet bases are a particular form of involutive bases which in turn are Gröbner bases with additional combinatorial properties which makes them very useful for many computations. They combine the classical Janet-Riquier theory of differential equations with the theory of Gröbner bases of polynomial ideals. Although Riquier and Janet developed originally their theory for nonlinear systems of differential equations, one obtains an algorithmic theory in the modern sense only for linear systems. Consequently, the author presents the theory of Janet bases in the context of Ore algebras. This class of non-commutative polynomial rings contains not only linear differential operators, but also linear difference or shift operators.

A Thomas decomposition deals with the problems posed by fully nonlinear systems where one usually must make case distictions depending on the initials and separants of the equations. As the case distinctions automatically lead to inequations, the decomposition is explained for systems composed of equations and inequations. First, the case of algebraic systems is treated; then the theory is extended to differential systems. Complete algorithms are exhibited and their implementation is discussed.

A number of different elimination problems are treated in the book. The first one is the classical algebraic problem of computing elimination ideals. It is considered here for ideals in an Ore algebra. In principle, this problem is solved by a Gröbner basis for the lexicographic term order. In practice, this approach is often not feasible within a reasonable amount of computing time. The author presents a technique which he calls degree steering and which consists of computing several Janet bases for different term orders. As another problem, the author also considers the question of “eliminating” standard basis vectors instead of variables for submodules of free modules. Also the classical task of finding the compatibility conditions for inhomogeneous systems of linear functional equations is treated here as an elimination problem.

The next type of problems has been term “linear differential elimination” by the author. Here one is given a family of complex analytic functions of the form \(\sum f_i(\alpha_i(z))g_i(z)\), where \(\alpha_i\) and \(g_i\) are prescribed and \(f_i\) arbitrary analytic functions. The task consists of deciding whether a further given analytic function \(u(z)\) is a member of this family and if yes to find corresponding parameters \(f_i\). The author solves this problem by obtaining an implicit representation of the family, namely a system of partial differential equations such that the family defines its solution space. Some applications to the construction of special solutions of system of differential equations are given.

Finally, a nonlinear version of this problem is treated where the given family of analytic functions is no longer a linear combination but of the general form \(p(f_1(\alpha_1(z),\dots,f_k(\alpha_k(z)))\). Again the basic idea is to obtain an implicit representation as solution space of a differential system. However, the nonlinearity makes it considerably harder to construct such a representation.

The second chapter on the used algorithmic tools mainly compiles and reviews results from the literature. But as in particular the Thomas decomposition is hardly known even to specialists in computer algebra, it is very useful to have this material presented in a uniform and succinct manner. The third chapter on elimination problems contains original results by the author. The book is well written with many pointers to the literature, in particular for those aspects that could not be covered in more details because of lack of space. All algorithms described in it have been implemented by the author and his collaborators in the computer algebra system Maple. For most of the packages, web addresses are given in the bibliography.

Reviewer: Werner M. Seiler (Kassel)

### MSC:

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

12H05 | Differential algebra |

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

16E05 | Syzygies, resolutions, complexes in associative algebras |

16S36 | Ordinary and skew polynomial rings and semigroup rings |

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

68W30 | Symbolic computation and algebraic computation |