Gröbner Bases pp 279-344 | Cite as

Gröbner Basis for Rings of Differential Operators and Applications

  • Nobuki Takayama


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.


Weight Vector Left Ideal Hypergeometric Series Singular Locus Hilbert Function 
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Copyright information

© Springer Japan 2013

Authors and Affiliations

  1. 1.Department of Mathematics, Graduate School of ScienceKobe UniversityKobeJapan

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