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Gröbner Basis for Rings of Differential Operators and Applications

In: Gröbner Bases

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  • Nobuki Takayama

    (Kobe University, Department of Mathematics, Graduate School of Science)

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.

Suggested Citation

  • Nobuki Takayama, 2013. "Gröbner Basis for Rings of Differential Operators and Applications," Springer Books, in: Takayuki Hibi (ed.), Gröbner Bases, edition 127, chapter 0, pages 279-344, Springer.
    • Handle: RePEc:spr:sprchp:978-4-431-54574-3_6
    DOI: 10.1007/978-4-431-54574-3_6
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