Primitive polynomial remainder sequences in elimination theory
type: Item
instance of: scholarly article
title: Primitive polynomial remainder sequences in elimination theory
zbMATH Open document ID: 0845.12006
DOI: 10.1007/BF01225644
published in: Applicable Algebra in Engineering, Communication and Computing
publication date: 1995-01-31T00:00:00Z
review text: The sequence of contents involved in the primitive polynomial remainder sequence (pprs) used to compute the gcd of two multivariate polynomials provides information about their common zeros. Let \(R:= K[x_1, \dots, x_n ]\) be a ring of polynomials over a field \(K\). If \[ f= \sum^m_{r=0} q_r (x_1, \dots, x_{n-1}) x^r_n\in \mathbb{R}, \qquad q_m\neq 0, \] then \(\text{lc} (f):= q_m\), \(\text{deg} (f):= m\), \(\text{cont} (f):= \text{gcd} (q_0, \dots, q_m)\) and \(\text{pp} (f):= f/\text{cont} (f)\). If \(\text{cont} (f)=1\) then \(f\) is called primitive. For non-zero \(f,g\in \mathbb{R}\) the pseudo-division relation is \[ (\text{lc} (g))^d f= \text{pquo} (f,g) g+\text{prem} (f,g), \qquad \text{deg(prem} (f,g))< \text{deg} (g), \] where \(d:= \max\{ \text{deg} (f)- \text{deg} (g)+ 1,0\}\) and \(\text{deg} (0) :=-1\). Let \(f_1, f_2\in \mathbb{R}\) be primitive non-zero polynomials with \(\text{deg} (f_1)\geq \text{deg} (f_2)\). The pprs of \(f_1\), \(f_2\) is the sequence of non-zero polynomials \(f_1, f_2, \dots, f_k\in \mathbb{R}\) such that \(\text{pp(prem} (f_{i-2}, f_{i-1}))= f_i\), \(i=3, \dots, k\), \(\text{prem} (f_{k-1}, f_k)=0\), and \(f_k= \text{gcd} (f_i, f_{i+1})\). Let \(c_i:= \text{cont(prem} (f_{i-1}, f_i))\), \(i=2, \dots, k-1\), \(l_i:= \text{lc} (f_i )^{\text{deg} (f_{i-1})- \text{deg} (f_i)+1}\), \(i=2, \dots, k\). Then the elimination sequence of \(f_1\), \(f_2\) is \(e_2, \dots, e_{k-1}\in K[x_1, \dots, x_{n-1} ]\) such that \[ e_2:= p_2, \quad e_i:= {{p_i} \over {\text{gcd} (p_i, \prod^{i-1}_{r=2} e_r)}}, \qquad i=3, \dots, k-1, \] where \[ p_i:= \text{squarefree } \Biggl( {{\prod^i_{r=2} c_r} \over {\text{gcd} (\prod^i_{r=2} c_r, \prod^i_{r=2} l_r)}} \Biggr), \qquad i=2, \dots, k-1, \] and \(\text{squarefree} (f)\) denotes the squarefree form of \(f\). Define \(\text{po} (h_1, h_2):= \max \{m\in \mathbb{N} \mid (h^m_1|h_2)\}\), \(h_1, h_2\in \mathbb{R}\). Then Lemma 1. Let \(p\in K[x_1, \dots, x_{n-1} ]\) be irreducible and \(i\in \{2, \dot…
Mathematics Subject Classification ID: 12Y05
Mathematics Subject Classification ID: 68W30
Mathematics Subject Classification ID: 13P10
zbMATH DE Number: 716267
zbMATH Keywords: multivariate polynomials
zbMATH Keywords: elimination sequence
zbMATH Keywords: polynomial equations
author: M. Kalkbrener
reviewed by: Francis Wright
describes a project that uses: CASA
describes a project that uses: Maple
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 716267