Abstract
Given a multivariate polynomial F(x, y, ...,z), this paper deals with calculating the roots ofF w.r.t.x in terms of formal power series or fractional-power series iny, ...,z. If the problem is regular, i.e. the expansion point is not a singular point of a root, then the calculation is easy, and the irregular case is considered in this paper. We extend the generalized Hensel construction slightly so that it can be applied to the irregular case. This extension allows us to calculate the roots of bivariate polynomial F(x, y) in terms of Puiseux series iny. For multivariate polynomial F(x, y, ...,z), we consider expanding the roots into fractional-power series w.r.t. the total-degree ofy, ...,z, and the roots are expressed in terms of the roots of much simpler polynomials.
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References
O. Aberth, Iterative method for finding all zeros of a polynomial simultaneously. Math. Comp.,27 (1973), 339–344.
E. Durand, Solutions numeriques des equations algébriques Tom 1: equation du tupe F(x) = 0. Racines d’um Polnome Masson, Paris, 1960.
D. Duval, Rational Puiseux expansions. Compos. Math.,70 (1989), 119–154.
K.O. Geddes, S.R. Czapor and G. Labahn, Algorithms for Computer Algebra. Kluwer Academic Pub., Boston-Dordrecht-London, 1992.
K. Hensel, Theorie der algebraischen Zahlen. Teubner, Berlin, 1908.
M. Iri, Numerical computation (in Japanese). Asakura Shoten, Tokyo, 1981.
I.O. Kerner, Ein Gesamtschrittverfahren zur Berechnung der Nullstellen von Polnomen. Numer. Math.,8 (1966), 290–294.
H.T. Kung and J.F. Traub, All algebraic functions can be computed fast. J. ACM,25 (1978), 245–260.
M. Lauer, Computing by homomorphic images. Computer Algebra, Symbolic and Algebraic Computation (eds. B. Buchberger, G.E. Collins and R. Loos), Springer-Verlag, Wien-New York, 1982.
V.A. Puiseux, Recherches sur les functions algébriques. J. Math.,15 (1850), 365–480.
T. Sasaki, T. Kitamoto and F. Kako, Error analysis of power series roots of multivariate algebraic equation. Preprint of Univ. Tsukuba, March 1994, 30 pages.
T. Sasaki and M. Sasaki, A unified method for multivariate polynomial factorization. Japan J. Indust. Appl. Math.,10 (1993), 21–39.
K. Shiihara and T. Sasaki, Analytic continuation and Riemann surface determination of algebraic functions by computer. Japan J. Indust. Appl. Math.,13 (1996), 107–116.
T. Sasaki, M. Suzuki, M. Kolář and M. Sasaki, Approximate factorization of multivariate polynomials and absolute irreducibility testing. Japan J. Indus. Appl. Math.,8 (1991), 357–375.
R.J. Walker, Algebraic Curves. Springer-Verlag, New York-Heidelberg-Berlin, 1978.
B. Wall, Puiseux expansion and annotated bibliography. Technical Report, RISC-Linz Series No. 91-46.0, Oct. 1991.
P.S. Wang, Parallel p-adic constructions in the univariate polynomial factoring algorithm. Proc. 1979 MACSYMA Users’ Conference, MIT Lab. Comp. Sci., MIT Boston, 1979, 310–318.
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Work supported in part by Japanese Ministry of Education, Science and Culture under Grants 03558008 and 04245102.
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Sasaki, T., Kako, F. Solving multivariate algebraic equation by Hensel construction. Japan J. Indust. Appl. Math. 16, 257–285 (1999). https://doi.org/10.1007/BF03167329
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DOI: https://doi.org/10.1007/BF03167329