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TitleGr\"obner basis for rings of differential operators and applications.
Author(s) Nobuki Takayama
TypeBook, Chapter in Book, Conference Proceeding
AbstractWe 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.
ISBN978-4-431-54573-6/hbk; 978-4-4
URL http://link.springer.com/chapter/10.1007%2F978-4-431-54574-3_6
PublisherTokyo: Springer
Translation No
Refereed No