Solving parametric piecewise polynomial systems
type: Item
instance of: scholarly article
title: Solving parametric piecewise polynomial systems
zbMATH Open document ID: 1251.65075
author: Yisheng Lai
author: Jinming Wu
author: Ren-Hong Wang
published in: Journal of Computational and Applied Mathematics
publication date: 2011-12-19T00:00:00Z
review text: This paper establishes the basic theory on piecewise polynomial functions that are \(C^r\) with different smoothness degree \(r\) over the whole domain, and presents the theory and methods for solving zero-dimensional parametric piecewise polynomial systems. It shows that solving such a system amounts to solving \(m\) parametric semi-algebraic systems, which is then reduced to the computation of \(m\) discriminant varieties. Here, \(m\) is the number of \(n\)-dimensional cells in the hereditary partition of the domain. The parametric piecewise polynomial system is ultimately solved utilizing the critical points method and the Collins partial cylindrical algebraic decomposition method. The paper also proposes a classification method and its algorithm to address whether there exists an open set in the parameter domain such that, for each point in the open set, the corresponding zero-dimensional non-parametric piecewise polynomial system (a realization of the parametric piecewise polynomial system) has exactly the given number of torsion-free real zeros in the \(m\) cells respectively.
Mathematics Subject Classification ID: 65H10
Mathematics Subject Classification ID: 14Q10
Mathematics Subject Classification ID: 65H05
Mathematics Subject Classification ID: 13P15
zbMATH DE Number: 5989526
zbMATH Keywords: piecewise polynomial
zbMATH Keywords: parametric piecewise polynomial system
zbMATH Keywords: parametric semi-algebraic systems
zbMATH Keywords: discriminant variety
zbMATH Keywords: number of real zeros
zbMATH Keywords: cylindrical algebraic decomposition method
zbMATH Keywords: algorithm
describes a project that uses: RAGlib
describes a project that uses: DISCOVERER
describes a project that uses: QEPCAD
MaRDI profile type: Publication
full work available at URL: j.cam.2011.05.008
OpenAlex ID: W2104612625
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DOI: 10.1016/J.CAM.2011.05.008
description: scientific article; zbMATH DE number 5989526