Nobuki Takayama

Grbner Deformations of Hypergeometric Differential Equations

The theory of Grbner bases is a main tool for dealing with rings of differential operators. This book reexamines the concept of Grbner bases from the point of view of geometric deformations. The algorithmic methods introduced in this book are particularly useful for studying the systems of multidimensional hypergeometric PDEs introduced by Gelfand, Kapranov, and Zelevinsky. A number of original research results are contained in the book, and many open problems are raised for future research in this rapidly growing area of computational mathematics.

Mathematical Software - ICMS 2006

By reading, you can know the knowledge and things more, not only about what you get from people to people. Book will be more trusted. As this mathematical software icms 2010 third international congress on mathematical software kobe japan september 13 17 2010 proceedings lecture notes in computer science, it will really give you the good idea to be successful. It is not only for you to be success in certain life you can be successful in everything. The success can be started by knowing the basic knowledge and do actions.

Algorithms for D-modules - restriction, tensor product, localization, and local cohomology groups

Abstract We consider an algebraic D -module M on the affine space, i.e. a system of linear partial differential equations with polynomial coefficients. We give an algorithm for computing the cohomology groups of the restriction of M to a linear subvariety by using a free resolution of M adapted to the V -filtration. Our algorithm works, at least, if M is holonomic. As applications, we obtain algorithms for computing tensor product, localization, and algebraic local cohomology groups of holonomic systems.

Gröbner basis and the problem of contiguous relations

It is a classical problem to find contiguous relations of hypergeometric functions of several variables. Recently Kametaka [11] and Okamoto [15] have developed the theory of hypergeometric solutions of the Toda equation. We need to find the explicit formulas of contiguous relations (or ladders) to construct the hypergeometric solutions of the Toda equation explicitly. We present an algorithm to obtain contiguous relations of hypergeometric functions of several variables. The algorithm is based on Buchberger’s algorithm [3] on the Gröbner basis.

An algorithm for de Rham cohomology groups of the complement of an affine variety via D-module computation

Abstract We give an algorithm to compute the following cohomology groups on U= C n ⧹V(f) for any non-zero polynomial f∈ Q [x 1 ,…,x n ] : 1. H k (U, C U ), C U is the constant sheaf on U with stalk C . 2. H k (U, V ), V is a locally constant sheaf of rank 1 on U. We also give partial results on computation of cohomology groups on U for a locally constant sheaf of general rank and on computation of H k ( C n ⧹Z, C ) where Z is a general algebraic set. Our algorithm is based on computations of Grobner bases in the ring of differential operators with polynomial coefficients.

Holonomic gradient descent and its application to the Fisher-Bingham integral

We give a new algorithm to find local maximum and minimum of a holonomic function and apply it for the Fisher-Bingham integral on the sphere

An approach to the zero recognition problem by Buchberger algorithm

S^n

An algorithm of constructing the integral of a module--an infinite dimensional analog of Gröbner basis

, which is used in the directional statistics. The method utilizes the theory and algorithms of holonomic systems.

Algebra,Geometry and Software Systems

Abstract The class of “holonomic function” is considered. We present a quasi-algorithm that recognizes whether a holonomic function is zero or not. The algorithm consists of procedures which obtain differential operators that annihilate sums, products, and definite integrals with, respect to parameters of holonomic functions. A Weyl algebra-analog of Buchbergers algorithm is used. A “holonomic” approach to the zero recognition algorithm was initiated by D. Zeilberger (1990) who realized it by Sylvesters dyalitic elimination. Our algorithm uses Buchbergers algorithm to improve Zeilbergers algorithm.

Gröbner basis, integration and transcendental functions

by introducing an analog of Grijbner basis of a submodule of a kind of infinite dimensional free module. We call M/an M the integral of the module AI. The non-commutativity of A,, prevents us from using the usual Buchberger algorithm to construct M/&M. (If A,, is commutative, then M/&M 21 An/(&, a>. There is no problem.) We must consider a sum of left and right ideal of A,. We overcome this difficulty by using an infinite dimensional analog of Grobner basis. The algorithm of constructing the integral of a module is not only important to mathematicians, but also has many impacts on the classical fields of computer algebra. It plays central roles in mathemat,ical formula verification [Zeil], [TakZ], computation of a definite integral [AZ], [Tak2] and an asymptotic expansion of a definite integral with respect to parameters. However, a complete algorithm of obtaining M/&M has not been known. We give a complete algorithm in this paper. The algorithm is an answer to the research problem of the paper [AZ]. We refer to [Buchl], [Buch2], [RIM], [FSK], [Bay] for the Griibner basis of a polynomial ideal and free module, to [Gal], [Gas], [Takl], [Nou] , [UT] for the Griibner basis of the ideal of Weyl algebra, to [Ber], [Bjo] for holonomic system and Weyl algebra. We remark that [Berg] also considered infinite set of reduction systems.