Advanced Studies in Pure Mathematics

Algorithms for $D$-modules, integration, and generalized functions with applications to statistics

Toshinori Oaku

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Abstract

This is an enlarged and revised version of the slides presented in a series of survey lectures given by the present author at MSJ SI 2015 in Osaka. The goal is to introduce an algorithm for computing a holonomic system of linear (ordinary or partial) differential equations for the integral of a holonomic function over the domain defined by polynomial inequalities. It applies to the cumulative function of a polynomial of several independent random variables with e.g., a normal distribution or a gamma distribution. Our method consists in Gröbner basis computation in the Weyl algebra, i.e., the ring of differential operators with polynomial coefficients. In the algorithm, generalized functions are inevitably involved even if the integrand is a usual function. Hence we need to make sure to what extent purely algebraic method of Gröbner basis applies to generalized functions which are based on real analysis.

Article information

Source
The 50th Anniversary of Gröbner Bases, T. Hibi, ed. (Tokyo: Mathematical Society of Japan, 2018), 253-352

Dates
Received: 30 August 2016
First available in Project Euclid: 21 September 2018

Permanent link to this document
https://projecteuclid.org/ euclid.aspm/1537499605

Digital Object Identifier
doi:10.2969/aspm/07710253

Mathematical Reviews number (MathSciNet)
MR3839713

Zentralblatt MATH identifier
07034256

Subjects
Primary: 13N10: Rings of differential operators and their modules [See also 16S32, 32C38] 13P10: Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) 46F10: Operations with distributions 62E15: Exact distribution theory

Keywords
$D$-module Gröbner basis generalized function probability density function

Citation

Oaku, Toshinori. Algorithms for $D$-modules, integration, and generalized functions with applications to statistics. The 50th Anniversary of Gröbner Bases, 253--352, Mathematical Society of Japan, Tokyo, Japan, 2018. doi:10.2969/aspm/07710253. https://projecteuclid.org/euclid.aspm/1537499605


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