Ritt-Wu characteristic set method for Laurent partial differential polynomial systems
type: Item
instance of: scholarly article
title: Ritt-Wu characteristic set method for Laurent partial differential polynomial systems
zbMATH Open document ID: 1470.12003
author: Youren Hu
author: Xiao-Shan Gao
published in: Journal of Systems Science and Complexity
publication date: 2019-03-06T00:00:00Z
review text: Let \(F\) be a field with \(m\) commuting derivations \(\{\delta_1, \dots, \delta_m\}\). Let \(\mathbb Y=\{y_1, \dots, y_n\}\) be differential indeterminates and \(F\{\mathbb Y\}\) the resulting differential ring. Let \(\Theta\) be the set of monic monomials in the \(\delta_i\) and let \(\Theta(\mathbb Y)\) be the set of all \(\theta y_j\) for \(\theta \in \Theta\) and \(1 \leq j \leq n\). Assume that \(\Theta(\mathbb Y)\) has been given a total order such that for any \(\alpha\), \(\beta\) and any \(i\) \(\delta_i(\alpha) \alpha\) and if \(\alpha \beta\) then \(\delta_i(\alpha) \delta_i(\beta)\). For \(f \in F\{\mathbb Y\}\), the leader of \(f\), \(\text{ld}_f\), is the largest element of \(\Theta(\mathbb Y)\) occurring in \(f\). If also \(g \in F\{\mathbb Y\}\), no proper derivative of \(\text{ld}_g\) occurs in \(f\), and as polynomials in \(\text{ld}_f\) (respectively \(\text{ld}_g\)) \(f\) has smaller degree then \(f\) is reduced with respect to \(g\). If \(S\) is a subset of \(F\{\mathbb Y\}\) then \(f\) is reduced with respect to \(S\) is it is reduced with respect to every element of \(S\). \(S\) is autoreduced if each element is reduced with respect to the set of the rest of the elements of \(S\). A differential chain is an autoreduced (finite) set whose elements are strictly increasing in rank. A polynomial \(f \in F\{\mathbb Y\}\) is monomial-primitive if it is not a multiple of any element of \(\Theta(\mathbb Y)\). The Laurent differential polynomial ring \(F\{\mathbb Y^\pm\}\) is the polynomial ring in the set of all \(\theta y_j, (\theta y_j)^{-1}\), regarded as a differential ring with the \(\delta_i\) extended by the quotient rule. Any \(f \in F\{\mathbb Y^\pm\}\) has an associated monomial-primitive normal form. A Laurent differential chain is a differential chain in \(F\{\mathbb Y\}\) all of who members are monomial-primitive. Let \(\mathbb P\) be a subset of \(F\{\mathbb Y^\pm\}\). A characteristic set for \(\mathbb P\) is a Laurent differential ch…
reviewed by: Andy R. Magid
Mathematics Subject Classification ID: 12H05
zbMATH DE Number: 7032428
zbMATH Keywords: Newton polygon
zbMATH Keywords: Laurent partial differential polynomial system
zbMATH Keywords: Laurent regular triangular set
zbMATH Keywords: Ritt-Wu characteristic set
MaRDI profile type: Publication
full work available at URL: s11424-019-8343-5
OpenAlex ID: W2914928059
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Wikidata QID: Q128377596
DOI: 10.1007/S11424-019-8343-5
description: scientific article; zbMATH DE number 7032428