A note on Gröbner bases and integration of rational functions
type: Item
instance of: scholarly article
title: A note on Gröbner bases and integration of rational functions
zbMATH Open document ID: 0844.13017
author: Günter Czichowski
published in: Journal of Symbolic Computation
publication date: 1996-08-27T00:00:00Z
review text: Given a rational function \(A(x)/D(x)\), let \(R(z)\) denote the resultant of \(A(x) - zD' (x)\) and \(D(x)\), regarded as polynomials in \(x\). Using an algorithm due to \textit{B. M. Trager} [in: Symbolic and algebraic computation, Proc. 1976 ACM Symp., Yorktown Heights/NY, 219-226 (1976; Zbl 0498.12005); also see \textit{D. Lazard} and \textit{R. Rioboo}, J. Symb. Comput. 9, No. 2, 113-115 (1990; Zbl 0723.68053)], the logarithmic part of the integral of \(A(x)/D(z)\) can be found by using the remainders that occur in the calculation of \(R(z)\) by the subresultant algorithm. The author presents an alternative way to obtain the logarithmic part by using a Gröbner basis. If \(I\) is the ideal generated by \(A(x) - zD' (x)\) and \(D(x)\), then the author shows that one can easily find the logarithmic part of the integral from the reduced Gröbner basis for \(I\) with respect to the lexicographic order with \(x > z\). The proof uses the fact that \(I\) is a zero-dimensional ideal whose zeros have distinct \(x\)-coordinates. No claim is made that this method is superior, from a computational standpoint, to Trager's algorithm.
Mathematics Subject Classification ID: 13P10
Mathematics Subject Classification ID: 68W30
zbMATH DE Number: 871128
zbMATH Keywords: integration of rational functions
zbMATH Keywords: Gröbner basis
reviewed by: Robert F. Lax
MaRDI profile type: Publication
full work available at URL: jsco.1995.1043
OpenAlex ID: W1991195948
DOI: 10.1006/JSCO.1995.1043
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description: scientific article; zbMATH DE number 871128