Gröbner deformations of hypergeometric differential equations.

*(English)*Zbl 0946.13021
Algorithms and Computation in Mathematics. 6. Berlin: Springer. viii, 254 p. (2000).

In recent years the theory of Gröbner bases has found several applications in various fields of symbolic computations, in particular in applications related to combinatorics. The second author of the present book has opened the view for further applications in combinatorics [see B. Sturmfels, “Gröbner bases and convex polytopes”, Univ. Lect. Ser. 8 (Providence 1996; Zbl 0856.13020)].

The present book offers another application of Gröbner bases related to new algorithms for dealing with rings of differential operators. Here the Gröbner bases are reexamined from the point of geometric deformations. More precisely it provides symbolic algorithms for constructing holomorphic solutions to systems of linear partial differential equations with polynomial coefficients. Such a system is represented by a left ideal \(I\) in the Weyl algebra \(D = {\mathbb C} \langle x_1, \ldots, x_n, \partial_1, \ldots, \partial_n \rangle.\) A Gröbner deformation of the left ideal \(I\) is the initial ideal \(\text{in}_{(-w,w)}(I) \subset D\) with respect to some generic weight vector \(w = (w_1, \ldots, w_n)\) with real coordinates \(w_i.\) By methods from computational commutative algebra there is an explicit solution basis for the Gröbner deformation \(\text{in}_{(-w,w)}(I).\) The problem is to extend it to a solution basis for \(I.\) This is solved under the natural hypothesis that the given \(D\)-ideal \(I\) is regular holonomic. This is valid, for instance, for the \(D\)-ideals representing hypergeometeric integrals.

The first chapter of the book is devoted to the basic notations. The classical Gauss’ hypergeometric function is expressed in the Gel’fand-Kapranov-Zelevinsky (GKZ) scheme. It contains also an introduction to holonomic systems of differential equations from the Gröbner basis point of view.

The second chapter is concerned with solving regular holonomic systems. The holomorphic solutions of \(I\) around a generic point in \({\mathbb C}^n\) form a vector space. If \(I\) is holonomic it is of finite dimension \(\text{rank} (I).\) The terms of such a holomorphic solution \(f\) are partially ordered by means of the generic weight vector \(w.\) The sum of the smallest terms, \(\text{in}_w(f),\) is a solution to the Gröbner deformation \(\text{in}_{(-w,w)} (I).\) The strategy of the solution is to find \(\text{in}_w(f)\) first and to construct \(f\) from it. This approach is justified by the inequality \(\text{rank in}_{(-w,w)} (I) \leq \text{rank} (I).\) It has to be an equality, which is true under certain circumstances. Moreover it is shown by an algorithmic approach that the equality holds whenever the system is regular holonomic. The main combinatorical arguments of the proofs grow out of the fact that the initial \(D\)-ideal \(\text{in}_{(-w,w)}(I)\) is torus-fixed for a generic \(w.\)

The chapter 3 is concerned with the GKZ-hypergeometric system \(H_A(\beta)\) associated with an integer matrix \(A\) and a complex parameter vector \(\beta,\) introduced by I. M. Gel’fand, A. V. Zelevinsky and M. M. Kapranov [Funct. Anal. Appl. 23, No. 2, 94-106 (1989); translation from Funkts. Anal. Prilozh. 23, No. 2, 12-26 (1989; Zbl 0787.33012)]. The matrix \(A\) represents a toric variety. If the variety lies in a projective space, then \(H_A(\beta)\) is regular holonomic, which is assumed throughout. The degree of the variety is called \(\text{vol}(A),\) since it coincides with the volume of the polytope spanned by \(A.\) By means of explicit deformations in the space of parameters the fundamental inequality \( \text{rank}(H_A(\beta)) \geq \text{vol}(A)\) is shown.

In chapter 4 the authors investigate when equality in the last estimate is true. Originally this was claimed to be true in general by Gel’fand, Zelevinsky and Kapranov [loc. cit.], which does not hold. Here the authors investigate three sufficient conditions for the equality. The first one is the Cohen-Macaulayness of a certain toric ideal \(I_A,\) the second, the so-called semi-resonance of the parameter \(\beta,\) and the third, the \(w\)-flatness of the parameter \(\beta,\) which depends on the Gröbner deformation. Moreover there is also an upper bound for the rank in general and an exact formula in the case of dimension 2.

The chapter 5 is devoted to the integration of \(D\)-modules. More precisely, hypergeometric functions arise naturally from integrals. It is the authors’ goal to present algorithms for computing asymptotic expansions of these kind of integrals by the following steps: First they compute the \(D\)-ideal consisting of all operators that annihilate the integrand. Secondly they find the annihilators of the integral via the machinery of \(D\)-module theoretic integration. Thirdly they compute the Nilson series expansion of the integral. Besides of this, the chapter serves also as a more general introduction to algorithms in algebraic geometry based on \(D\)-modules. The GKZ system remains the focus example for all the general concepts and constructions in the same spirit as toric varieties serve as a ubiquitous source of examples in algebraic geometry.

In an appendix there is a description of current computer systems for \(D\)-modules and their design. This completes the algorithmic picture of the whole book with concrete samples of computations. The book is well written. The project for the book started when the authors came together to work on joint research on topics now contained in chapter 4. Then they started to develop all the necessary basic material about \(D\)-modules and linear partial differential equations not available in the literature.

The monograph requires a consequent reading in order to discover all the beauties and the surprising connections between several different branches of mathematics, coming together in the text. This book contains a number of original research results on holonomic systems and hypergeometric functions. The reviewer is sure that it will be the standard reference for computational aspects and research on \(D\)-modules in the future. It raises many open problems for future work in this area.

The present book offers another application of Gröbner bases related to new algorithms for dealing with rings of differential operators. Here the Gröbner bases are reexamined from the point of geometric deformations. More precisely it provides symbolic algorithms for constructing holomorphic solutions to systems of linear partial differential equations with polynomial coefficients. Such a system is represented by a left ideal \(I\) in the Weyl algebra \(D = {\mathbb C} \langle x_1, \ldots, x_n, \partial_1, \ldots, \partial_n \rangle.\) A Gröbner deformation of the left ideal \(I\) is the initial ideal \(\text{in}_{(-w,w)}(I) \subset D\) with respect to some generic weight vector \(w = (w_1, \ldots, w_n)\) with real coordinates \(w_i.\) By methods from computational commutative algebra there is an explicit solution basis for the Gröbner deformation \(\text{in}_{(-w,w)}(I).\) The problem is to extend it to a solution basis for \(I.\) This is solved under the natural hypothesis that the given \(D\)-ideal \(I\) is regular holonomic. This is valid, for instance, for the \(D\)-ideals representing hypergeometeric integrals.

The first chapter of the book is devoted to the basic notations. The classical Gauss’ hypergeometric function is expressed in the Gel’fand-Kapranov-Zelevinsky (GKZ) scheme. It contains also an introduction to holonomic systems of differential equations from the Gröbner basis point of view.

The second chapter is concerned with solving regular holonomic systems. The holomorphic solutions of \(I\) around a generic point in \({\mathbb C}^n\) form a vector space. If \(I\) is holonomic it is of finite dimension \(\text{rank} (I).\) The terms of such a holomorphic solution \(f\) are partially ordered by means of the generic weight vector \(w.\) The sum of the smallest terms, \(\text{in}_w(f),\) is a solution to the Gröbner deformation \(\text{in}_{(-w,w)} (I).\) The strategy of the solution is to find \(\text{in}_w(f)\) first and to construct \(f\) from it. This approach is justified by the inequality \(\text{rank in}_{(-w,w)} (I) \leq \text{rank} (I).\) It has to be an equality, which is true under certain circumstances. Moreover it is shown by an algorithmic approach that the equality holds whenever the system is regular holonomic. The main combinatorical arguments of the proofs grow out of the fact that the initial \(D\)-ideal \(\text{in}_{(-w,w)}(I)\) is torus-fixed for a generic \(w.\)

The chapter 3 is concerned with the GKZ-hypergeometric system \(H_A(\beta)\) associated with an integer matrix \(A\) and a complex parameter vector \(\beta,\) introduced by I. M. Gel’fand, A. V. Zelevinsky and M. M. Kapranov [Funct. Anal. Appl. 23, No. 2, 94-106 (1989); translation from Funkts. Anal. Prilozh. 23, No. 2, 12-26 (1989; Zbl 0787.33012)]. The matrix \(A\) represents a toric variety. If the variety lies in a projective space, then \(H_A(\beta)\) is regular holonomic, which is assumed throughout. The degree of the variety is called \(\text{vol}(A),\) since it coincides with the volume of the polytope spanned by \(A.\) By means of explicit deformations in the space of parameters the fundamental inequality \( \text{rank}(H_A(\beta)) \geq \text{vol}(A)\) is shown.

In chapter 4 the authors investigate when equality in the last estimate is true. Originally this was claimed to be true in general by Gel’fand, Zelevinsky and Kapranov [loc. cit.], which does not hold. Here the authors investigate three sufficient conditions for the equality. The first one is the Cohen-Macaulayness of a certain toric ideal \(I_A,\) the second, the so-called semi-resonance of the parameter \(\beta,\) and the third, the \(w\)-flatness of the parameter \(\beta,\) which depends on the Gröbner deformation. Moreover there is also an upper bound for the rank in general and an exact formula in the case of dimension 2.

The chapter 5 is devoted to the integration of \(D\)-modules. More precisely, hypergeometric functions arise naturally from integrals. It is the authors’ goal to present algorithms for computing asymptotic expansions of these kind of integrals by the following steps: First they compute the \(D\)-ideal consisting of all operators that annihilate the integrand. Secondly they find the annihilators of the integral via the machinery of \(D\)-module theoretic integration. Thirdly they compute the Nilson series expansion of the integral. Besides of this, the chapter serves also as a more general introduction to algorithms in algebraic geometry based on \(D\)-modules. The GKZ system remains the focus example for all the general concepts and constructions in the same spirit as toric varieties serve as a ubiquitous source of examples in algebraic geometry.

In an appendix there is a description of current computer systems for \(D\)-modules and their design. This completes the algorithmic picture of the whole book with concrete samples of computations. The book is well written. The project for the book started when the authors came together to work on joint research on topics now contained in chapter 4. Then they started to develop all the necessary basic material about \(D\)-modules and linear partial differential equations not available in the literature.

The monograph requires a consequent reading in order to discover all the beauties and the surprising connections between several different branches of mathematics, coming together in the text. This book contains a number of original research results on holonomic systems and hypergeometric functions. The reviewer is sure that it will be the standard reference for computational aspects and research on \(D\)-modules in the future. It raises many open problems for future work in this area.

Reviewer: Peter Schenzel (Halle)

##### MSC:

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

68W30 | Symbolic computation and algebraic computation |

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

68-02 | Research exposition (monographs, survey articles) pertaining to computer science |

33D60 | Basic hypergeometric integrals and functions defined by them |

13N10 | Commutative rings of differential operators and their modules |

14M25 | Toric varieties, Newton polyhedra, Okounkov bodies |

16S32 | Rings of differential operators (associative algebraic aspects) |

14F10 | Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials |