×

Specialization of the Sturm sequence. (Spécialisation de la suite de Sturm.) (French) Zbl 0999.12502

The existing algorithms for computing the number of real roots of a polynomial and their extensions are considered. Links between these methods and the Sturm-Habicht sequence as well as between this sequence and subresultant theory are stated.

MSC:

12Y05 Computational aspects of field theory and polynomials (MSC2010)
68W30 Symbolic computation and algebraic computation
12D10 Polynomials in real and complex fields: location of zeros (algebraic theorems)
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] FROBENIUS, Uber das Traegheitsgesetz des quadratishen Formen, S-B Pruss. Akad. Wiss., Marz 1984, 241-256; Mai 1984, 403-431. JFM25.0318.01 · JFM 25.0318.01
[2] Fr. GANTMACHER, Théorie des matrices, t. I, Dunod 1966. Zbl0136.00410 · Zbl 0136.00410
[3] L. GONZALEZ, H. LOMBARDI, T. RECIO, M.-F. ROY, Spécialisation de la suite de Sturm et sous-résultants (I), Inf. Théorique et Appl., 1990, 6, 561-588. Zbl0732.68059 MR1082916 · Zbl 0732.68059
[4] L. GONZALEZ, H. LOMBARDI, T. RECIO et M.-F. ROY, Sturm-Habicht sequences, Proceedings ISSAC, 1989.
[5] L. GONZALEZ, H. LOMBARDI, T. RECIO, M.-F. ROY, Spécialisation de la suite de Sturm et sous-résultants, Version détaillée, dans CALSYF (journées du GRECO de Calcul Formel) 1989.
[6] L. GONZÁLEZ-VEGA, H. LOMBARDI, T. RECIO, M.-F. ROY, Determinants and real roots of univariate polynomials. A paraître dans : Special volume of the series ”Texts and Monographs in Symbolic Computation” (Springer-Verlag) ayant pour titre : 25 years of Quantifier Elimination and Cylindrical Algebraic Decomposition, (Compte-rendus du Symposium on quantifier elimination andcylindrical algebraic decomposition. Linz. 6-8 oct. 93). Zbl0900.12002 · Zbl 0900.12002
[7] L. GONZALEZ, The Proof of the Sylvester Theorem Through Habicht Sequences, Preprint, Université de Santander, 1988.
[8] W. HABICHT, Eine Verallgemeinerung des Sturmschen Wurzel zählverfahrens, Comm. Math. Helvetici, 1948, 21, 99-116. Zbl0029.24402 MR23796 · Zbl 0029.24402
[9] C. HERMITE, Remarques sur le théorème de Sturm, C.R. Acad. Sci. Paris, 1853, 36, 52-54.
[10] M. G. KREIN, M. A. NAIMARK, The Method of Symmetric and Hermitian Forms on the Theory of the Separation of the Roots of Algebraic Equations, Originairement publié à Kharkov (1936), Lin. Multilinear algebra, 1981, 10, 265-308. Zbl0584.12018 MR638124 · Zbl 0584.12018
[11] J. J. SYLVESTER, On a Theory of Syzygetic Relations of Two Rational Intégral Functions, Comprising an Application to the Theory of Sturm’s Function, Trans. Roy. Soc. London, 1853. Reprint dans : Sylvester, Collected Math. Papers, Chelsea Pub. Comp. NY, 1983, vol. 1, 429-586.
[12] A. VALLIBOUZE, Fonctions symétriques et changements de base, Thèse, Université Paris-VI, 1987.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.