On factoring parametric multivariate polynomials
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.
scientific article; zbMATH DE number 5793903
scientific article