1 edition of **Gröbner Deformations of Hypergeometric Differential Equations** found in the catalog.

- 170 Want to read
- 37 Currently reading

Published
**2000**
by Springer Berlin Heidelberg in Berlin, Heidelberg
.

Written in English

- Combinatorics,
- Mathematics,
- Global analysis (Mathematics),
- Algorithms,
- Geometry, algebraic

In recent years, new algorithms for dealing with rings of differential operators have been discovered and implemented. A main tool is the theory of Gröbner bases, which is reexamined here from the point of view of geometric deformations. Perturbation techniques have a long tradition in analysis; Gröbner deformations of left ideals in the Weyl algebra are the algebraic analogue to classical perturbation techniques. The algorithmic methods introduced in this book are particularly useful for studying the systems of multidimensional hypergeometric partial differentiel equations introduced by Gel"fand, Kapranov and Zelevinsky. The Gröbner deformation of these GKZ hypergeometric systems reduces problems concerning hypergeometric functions to questions about commutative monomial ideals, and thus leads to an unexpected interplay between analysis and combinatorics. This book contains a number of original research results on holonomic systems and hypergeometric functions, and it raises many open problems for future research in this rapidly growing area of computational mathematics "

**Edition Notes**

Statement | by Mutsumi Saito, Bernd Sturmfels, Nobuki Takayama |

Series | Algorithms and Computation in Mathematics -- 6, Algorithms and Computation in Mathematics -- 6 |

Contributions | Sturmfels, Bernd, Takayama, Nobuki |

Classifications | |
---|---|

LC Classifications | QA299.6-433 |

The Physical Object | |

Format | [electronic resource] / |

Pagination | 1 online resource (viii, 254 p.) |

Number of Pages | 254 |

ID Numbers | |

Open Library | OL27041791M |

ISBN 10 | 3642085342, 366204112X |

ISBN 10 | 9783642085345, 9783662041123 |

OCLC/WorldCa | 851392646 |

on bivariate hypergeometric functions and how it motivates the deﬁnition of GKZ systems. 1. Hypergeometric Series and Differential Equations The Gamma Function and the Pochhammer Symbol. We recall that the Gamma function Γ(s) may be deﬁned by the integral: () Γ(s) = . 4 CHAPTER 1. ORDINARY LINEAR DIFFERENTIAL EQUATIONS Note that if we replace y by Sy in the system, where S ∈ GL(n,K), we obtain a new system for the new y, ∂y = (S−1AS +S−1∂S)y. Two n×n-systems with coeﬃcient matrices A,B are called equivalent over K if there exists S ∈ GL(n,K) such that B = S−1AS +S−1∂S. It is well known that a diﬀerential system can be rewritten as a File Size: KB.

In the following we solve the second-order differential equation called the hypergeometric differential equation using Frobenius method, named after Ferdinand Georg Frobenius. This is a method that uses the series solution for a differential equation, where we assume the solution takes the form of a series. The origins of the mathematics in this book date back more than two thou sand years, as can be seen from the fact that one of the most important algorithms presented here bears the name of the Greek mathematician Eu clid. The word "algorithm" as well as the key word "algebra" in the title of this book come from the name and the work of the ninth-century scientist Mohammed ibn Musa al.

The Newton polygon construction for ODEs, and Malgrange–Ramis polygon for partial differential equations in one variable are generalized in order to g Cited by: e solutions of hypergeometric di erential equation include many of the most interesting special functions of mathematical physics. Solutions to the hypergeometric dif-ferential equation are built out of the hypergeometric series. e solution of Euler s hypergeometric di erential equation is called hypergeometric function or Gaussian function 2 1.

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Perturbation techniques have a long tradition in analysis; Gröbner deformations of left ideals in the Weyl algebra are the algebraic analogue to classical perturbation techniques. The algorithmic methods introduced in this book are particularly useful for studying the systems of multidimensional hypergeometric partial differentiel equations introduced by Gel'fand, Kapranov and by: Perturbation techniques have a long tradition in analysis; Gröbner deformations of left ideals in the Weyl algebra are the algebraic analogue to classical perturbation techniques.

The algorithmic methods introduced in this book are particularly useful for studying the systems of multidimensional hypergeometric partial differentiel equations introduced by Gel'fand, Kapranov and Zelevinsky.

The theory of Gröbner bases is a main tool for dealing with rings of differential operators. This book reexamines the concept of Gröbner bases from the point of view of geometric deformations. The theory of Groebner bases is a main tool for dealing with rings of differential operators.

This book reexamines the concept of Groebner 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 PDE's introduced by Gelfand, Kapranov, and Zelevinsky.

Nobuki Takayama is a Professor at Kobe University, Japan. His research fields comprise computer algebra, hypergeometric functions, D-modules, and algebraic statistics. He co-authored “Gröbner Deformations of Hypergeometric Differential Equations” (), published by Springer.

Table of contents (9 chapters). Cite this chapter as: Saito M., Sturmfels B., Takayama N. () Hypergeometric Series. In: Gröbner Deformations of Hypergeometric Differential Equations.

Gröbner deformations of hypergeometric differential equations By Mutsumi Saito, Bernd Sturmfels Gröbner Deformations of Hypergeometric Differential Equations book Nobuki Takayama No static citation data No static citation data Cite. It is proved that for a system of linear partial differential equations with polynomial coefficients, the Gröbner basis in the Weyl algebra is sufficient for the computation of the characteristic.

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 PDE's introduced by Gelfand, Kapranov, and Zelevinsky. DOI: /kyushujm Corpus ID: GRÖBNER BASIS AND SINGULAR LOCUS OF LAURICELLA'S HYPERGEOMETRIC DIFFERENTIAL EQUATIONS @inproceedings{NakayamaGRBNERBA, title={GR{\"O}BNER BASIS AND SINGULAR LOCUS OF LAURICELLA'S HYPERGEOMETRIC DIFFERENTIAL EQUATIONS}, author={Hiromasa Nakayama}.

els, and ma, Gröbner Deformations of Hypergeometric Differential Equations, Springer, How to calculate the slopes of a D-module Article. The solutions of hypergeometric differential equation include many of the most interesting special functions of mathematical physics.

Solutions to the hypergeometric differential equation are built out of the hypergeometric series. Definition 1.

The Pochhammer -symbol is Cited by: 8. COVID Resources. Reliable information about the coronavirus (COVID) is available from the World Health Organization (current situation, international travel).Numerous and frequently-updated resource results are available from this ’s WebJunction has pulled together information and resources to assist library staff as they consider how to handle coronavirus.

The generalization of this equation to three arbitrary regular singular points is given by Riemann's differential equation. Any second order differential equation with three regular singular points can be converted to the hypergeometric differential equation by a change of variables.

Solutions at the singular points. Solutions to the hypergeometric differential equation are built out of the hypergeometric series. Bernd Sturmfels (born Ma in Kassel, West Germany) is a Professor of Mathematics and Computer Science at the University of California, Berkeley and is a director of the Max Planck Institute for Mathematics in the Sciences in Leipzig since We study a hypergeometric function in two variables and a system of hypergeometric differential equations associated with this function.

This is a regular holonomic system of rank give a fundamental system of solutions to this system in terms of this hypergeometric by: 1. It is known that the A-hypergeometric function satisfies a system of linear partial differential equations in x, which is called the A-hypergeometric system.

The A -hypergeometric system is a holonomic system, and the operators of the system generate a zero-dimensional ideal in the ring of differential operators with rational function Cited by: 3. Nobuki Takayama: free download. Ebooks library. On-line books store on Z-Library | B–OK. Download books for free.

Find books Two Algebraic Byways From Differential Equations: Gröbner Bases and Quivers. Groebner Deformations of Hypergeometric Differential Equations. Springer. Mutsumi Saito, Bernd Sturmfels. Groebner basis and singular locus of Lauricella's hypergeometric differential equations Item Preview.

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Free shipping for many products!. Groebner bases for differential operators with field coefficients (reference request) Takayama, Grobner Deformations of Hypergeometric Differential Equations comes to mind.

Have you had a "Grobner Deformations of Hypergeometric Differential Equations", the case this book addresses is the case of the Weyl algebra over a characteristic 0.The hypergeometric and Legendre functions with applications to integral equations of potential theory Unknown Binding – January 1, by Chester Snow (Author)Author: Chester Snow.wise speci ed.

However, the hypergeometric function is not de ned if any b j;j= 1;;qare real and equal to a non-positive integer, and there are numerical issues in its computation if one or more values of b j are close to a non-positive integer.

The generalized hypergeometric function pF qis known to satisfy the following di erential.