Some remarks on Fitzpatrick and Flynn's Gröbner basis technique for Padé approximation.
type: Item
instance of: scholarly article
title: Some remarks on Fitzpatrick and Flynn's Gröbner basis technique for Padé approximation.
zbMATH Open document ID: 1050.41013
DOI: 10.1016/S0747-7171(03)00041-5
author: John B. Little
author: David Ortiz
author: Ricardo Ortiz-Rosado
author: Rebecca Pablo
author: Karen R. Ríos-Soto
published in: Journal of Symbolic Computation
publication date: 2003-08-19T00:00:00Z
review text: The authors continue work of \textit{P. Fitzpatrick} and \textit{J. Flynn} [J. Symb. Comput. 13, No. 2, 133--138 (1992; Zbl 0746.68045)] exploiting Gröbner bases for obtaining multivariable Pade-approximants. Consider \(h\in k[\underline{x}]=k[x_1,\ldots, x_s]\) representing an initial segment of a Taylor series of a function to be approximated. In the equation lattice approach to multivariate Padé approximation one prescribes finite subsets \(N,D\subset \mathbb Z_{\geq 0}^s\) and a third finite subset \(E\subset \mathbb Z_{\geq 0}^s\) and seeks to determine \(a(x)=\sum_{\alpha\in N} a_\alpha x^\alpha, \) \(b(x)=\sum_{\beta\in N} b_\beta x^\beta\) so that \(a(x)-b(x)h(x)= \sum_{\gamma\in \mathbb Z_{\geq 0}^s \setminus E } u_\gamma x^\gamma.\) To be well posed, one requires certain exponent conditions, like that \(| N| +| D| \geq | E| +1,\) and if \(\gamma \in E, \gamma=\delta+\varepsilon,\) then \(\delta,\epsilon \in E.\) Let \(I=\text{span}_k \{x^\gamma: \gamma\in \mathbb Z_{\geq 0}^s \setminus E \}.\) \(I\) is a monomial ideal in \(k[\underline{x}].\) The elements \((a,b)\) of \(M=\{(a,b)\in k[\underline{x}]^2: a\equiv bh \bmod I \}\) are considered as generalizations of classical Padé approximants. On the module \(k[\underline{x}]^2\) the authors define an order \(> \tau\) as follows: \(x^\alpha e_i >\tau x^\beta e_j\) iff \(| \alpha| >| \beta| \) or \(| \alpha| =| \beta| \) and \(i<j\) or \(| \alpha| =| \beta| \) and \(i=j\) and \(x^\alpha > x^\beta,\) the latter referring to a prefixed monomial order on \(k[\underline{x}]\) [see Ch. 5 of \textit{D. Cox}, \textit{J. Little} and \textit{D. O'Shea} [Using Algebraic Geometry (1998; Zbl 0920.13026) for concepts and notation]. The authors prove that if there exists an up to constant multiples unique \((a_0,b_0)\in M\) such that \(\tau(a_0)< \tau(b_0)\leq m\) then such must also appear in any Groebner basis for \(M\) w.r.t. the \(> \tau\) order. Here \(\tau(.)\) means total degree, and \(m\) is a bound on the degrees …
Mathematics Subject Classification ID: 41A21
Mathematics Subject Classification ID: 13P10
zbMATH DE Number: 1967112
zbMATH Keywords: multivariable Padé approximation
zbMATH Keywords: Groebner bases for modules
zbMATH Keywords: monomial order
reviewed by: Alexander Kovačec
MaRDI profile type: Publication
cites work: Q4305609
cites work: Q4209232
cites work: How well can the concept of Padé approximant be generalized to the multivariate case?
cites work: Solving a multivariable congruence by change of term order
cites work: A Gröbner basis technique for Padé approximation
cites work: Q4146054
full work available at URL: s0747-7171(03)00041-5
OpenAlex ID: W1963636946
description: scientific article; zbMATH DE number 1967112