On factoring parametric multivariate polynomials
type: Item
instance of: scholarly article
title: On factoring parametric multivariate polynomials
zbMATH Open document ID: 1200.11096
author: Ali Ayad
published in: Advances in Applied Mathematics
publication date: 2010-09-30T00:00:00Z
review text: The present paper gives an algorithm for the absolute factorization (factorization over the algebraic closure \(\bar{\mathbb{Q}}\)) of a parametric multivariate polynomial \(F\in \mathbb{Q}[u_1,\dots, u_r][X_0,\dots, X_n]\). The method is a follow-up, with some improvements, of a previous work of the author [Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. 316, 5--29 (2004); translation in J. Math. Sci. 134, 2325--2339 (2006; Zbl 1077.65046)]. Section 1 formalizes the notion of Parametric Absolute Factorization (PAF) of \(F\)\, and stated the main result (theorem 1.5): it is possible to find a finite number of PAFs whose constructible sets form a partition of the parameters space \(\mathcal{P}=\bar{\mathbb{Q}}^r\). Section 2 gathers some necessary tools: Hensel's lemma, the Chistov-Grigoriev method for quantifier elimination [\textit{A. L. Chistov} and \textit{D. Yu. Grigorie}, Mathematical foundations of computer science, Proc. 11th Symp., Praha/Czech. 1984, Lect. Notes Comput. Sci. 176, 17--31 (1984; Zbl 0562.03015)] and an algorithm of the author of the present paper for solving zero-dimensional parametric polynomial systems [PhD Thesis, University of Rennes 1, France (2006)]. The proposed algorithm is detailed in the section 3. Theorem 3.19 includes Theorem 1.5 and gives also bounds for the number of necessary PAFs and the computational complexity of the algorithm.
Mathematics Subject Classification ID: 11Y16
Mathematics Subject Classification ID: 68W30
Mathematics Subject Classification ID: 11C08
Mathematics Subject Classification ID: 12D05
Mathematics Subject Classification ID: 14C05
zbMATH DE Number: 5793903
zbMATH Keywords: symbolic computation
zbMATH Keywords: complexity analysis
zbMATH Keywords: polynomial absolute factorization
zbMATH Keywords: parametric polynomials
reviewed by: Juan Tena Ayuso
MaRDI profile type: Publication
full work available at URL: j.aam.2010.03.006
OpenAlex ID: W2014017290
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DOI: 10.1016/J.AAM.2010.03.006
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description: scientific article; zbMATH DE number 5793903