A Gröbner basis technique for Padé approximation
type: Item
instance of: scholarly article
title: A Gröbner basis technique for Padé approximation
zbMATH Open document ID: 0746.68045
DOI: 10.1016/S0747-7171(08)80087-9
author: S. Singh
published in: Journal of Symbolic Computation
publication date: 1992-09-26T00:00:00Z
review text: Let \(a,b,h\in R=k[x_ 1,\dots,x_ n]\) and let \(I=(p_ 1,\dots,p_ r)\) be an ideal in \(R\) where the given basis is a Gröbner basis. The solutions \((a,b)\) of the congruence \(a\equiv bh \mod I\) are considered as generalizations of classical Padé approximants. One basis for the solution module \(M\) is \(\{(h,1),(Lt(p_ j):\;1\leq j\leq n\}\) where \(Lt(p)\) denotes the leading term of \(p\). Suppose a solution \((a,b)\) exists with \(a\) and \(b\) relatively prime and in normal form modulo \(I\), and satisfying a weak term order condition: \(Lt(a)\leq\varphi\), \(Lt(b)\leq\psi\) and for all \(\rho\), \(\sigma\) with \(\rho\leq_ T\varphi\), \(\sigma\leq_ T\psi\) and \(\rho,\sigma\not\in(Lt(I))\) the product \(\rho\sigma\) does not lie in \((Lt(I))\). (Here \((Lt(I))\) is the ideal generated by the leading terms of the elements of \(I\), \(\varphi\), \(\psi\), \(\rho\), \(\sigma\) are terms and \(<_ T\) is a term order.) Then \((a,b)\) is a minimal (reduced) solution relative to the term order \(<_ w\) on \(R^ 2\) induced from \(<_ T\) and the weight vector \(w=(\psi,\varphi)\). A scalar multiple of \((a,b)\) appears in any Gröbner basis of \(M\) relative to \(<_ w\). Such a Gröbner basis can be derived from the one given above by changing the term order.
Mathematics Subject Classification ID: 68W30
Mathematics Subject Classification ID: 13P10
Mathematics Subject Classification ID: 41A21
zbMATH DE Number: 56191
zbMATH Keywords: Gröbner basis
zbMATH Keywords: Padé approximants
reviewed by: Patrick Fitzpatrick
MaRDI profile type: Publication
cites work: On the complexity of computing syzygies
cites work: Q3340031
cites work: Q3935355
cites work: Q3698165
cites work: New constructive methods in classical ideal theory
cites work: Extension of the Berlekamp-Massey algorithm to N dimensions
description: scientific article; zbMATH DE number 56191