Gröbner Basis for Rings of Differential Operators and Applications

Nobuki Takayama ()
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Nobuki Takayama: Kobe University, Department of Mathematics, Graduate School of Science

Chapter Chapter 6 in Gröbner Bases, 2013, pp 279-344 from Springer

Abstract: 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.

Keywords: Weight Vector; Left Ideal; Hypergeometric Series; Singular Locus; Hilbert Function (search for similar items in EconPapers)
Date: 2013
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-4-431-54574-3_6

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DOI: 10.1007/978-4-431-54574-3_6

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