Gröbner Basis for Rings of Differential Operators and Applications

Abstract

We introduce the theory and present some applications of Gröbner bases for the rings of differential operators with rational function coefficients R and for those with polynomial coefficients D.The discussion with R, in the first half, is elementary. In the ring of polynomials, zero-dimensional ideals form the biggest class, and this is also true in R. However, in D, there is no zero-dimensional ideal, and holonomic ideals form the biggest class. Most algorithms for D use holonomic ideals.As an application, we present an algorithm for finding local minimums of holonomic functions; it can be applied to the maximum-likelihood estimate. The last part of this chapter considers A-hypergeometric systems; topics covered in other chapters will reappear in the study of A-hypergeometric systems. We have provided many of the proofs, but some technical proofs in the second half of this chapter have been omitted; these may be found in the references at the end of this chapter.

Special values of Gauss's hypergeometric series derived from Appell's series $F_1$ with closed forms

“…The relation (2.1) is called the "contiguity relation" for F 1 (or the "four-term relation" for F 1 ). Note that it is possible to compute Q 10 , Q 01 and Q 00 exactly by using the method introduced in [Ta1] (see also [Ta2]). Next, we define…”

Algebraic analysis of rotation data

“…We now address this by introducing dimensionality reduction via invariant theory. The same approach makes sense for the general definition of J seen in (30).…”

“…In this situation, we can associate a Pfaffian system to the D-ideal I . For the general theory, we refer the reader to [30] and specifically to [28, (23)].…”

Special Values of the Hypergeometric Series

“…This relation is called the contiguity relation of 3 F 2 (or the four term relation of example, see [Ta1]. From (A.1), we have 3 F 2 a 1 + nk 1 , a 2 + nk 2 , a 3 + nk 3 b 1 + nl 1 , b 2 + nl 2 ; x = Q (n) 2 (x) 3 F 2 a 1 + (n − 1)k 1 + 2, a 2 + (n − 1)k 2 + 2, a 3 + (n − 1)k 3 + 2 b 1 + (n − 1)l 1 + 2, b 2 + (n − 1)l 2 + 2 ; x + Q (n) 1 (x) 3 F 2 a 1 + (n − 1)k 1 + 1, a 2 + (n − 1)k 2 + 1, a 3 + (n − 1)k 3 + 1 b 1 + (n − 1)l 1 + 1, b 2 + (n − 1)l 2 + 1 ; x + Q (n) 0 (x) 3 F 2 a 1 + (n − 1)k 1 , a 2 + (n − 1)k 2 , a 3 + (n − 1)k 3 b 1 + (n − 1)l 1 , b 2 + (n − 1)l 2 ; x ,…”

“…This relation is called the contiguity relation of F 1 (or the four term relation of F 1 ). As to a calculation method of coefficients for the above relation, for example, see [Ta1] and [Ta2]. We define Q Then, for this sextuple, we have the first order difference equation Solving (A.14) or (A.15) as the final step, we will find hypergeometric identity for F 1 .…”