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Multi-variate Polynomials and Newton-Puiseux Expansions

  • Frédéric Beringer
  • Françoise Richard-Jung
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2630)

Abstract

The classical Newton polygon method for the local resolution of algebraic equations in one variable can be extended to the case of multi-variate equations of the type f(y) = 0, f ∈ ℂ [x 1,..., x N ][y]. For this purpose we will use a generalization of the Newton polygon - the Newton polyhedron - and a generalization of Puiseux series for several variables.

Keywords

Series Solution Newton Algorithm Newton Polygon Newton Polyhedron Edge Path 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Brieskorn, E., Knörrer, H. (1986): Plane algebraic curves. Birkhäuser VerlagGoogle Scholar
  2. 2.
    Della-Dora, J., Richard-Jung, F. (1997): About the newton polygon algorithm for non linear ordinary differential equations. Proceedings of the International symposium on symbolic and algebraic computationGoogle Scholar
  3. 3.
    Walker, R. J. (1950): Algebraic curves. Dover editionGoogle Scholar
  4. 4.
    Beringer, F., Jung, F. (1998) Solving “Generalized Algebraic Equations”. Proceedings of the International symposium on symbolic and algebraic computationGoogle Scholar
  5. 5.
    McDonald, J. (1995): Fiber polytopes and fractional power series. Journal of Pure and applied Alebra, 104, 213–233zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Ewald, G. Combinatorial Convexity and Algebraic Geometry. Graduate Texts in Mathematics, SpringerGoogle Scholar
  7. 7.
    González Pérez, P.D. (2000): Singularités quasi-ordinaires toriques et polyèdre de Newton du discriminant. Canad. J. Math., 52, 348–368zbMATHMathSciNetGoogle Scholar
  8. 8.
    Alonso, M.E. Luengo, I., Raimondo, M. (1989): An algorithm on quasi-ordinary polynomials. Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, Lecture Notes in Computer Science., 357, 59–73Google Scholar
  9. 9.
    Oda, T. (1988): Convex bodies and algebraic geometry: an introduction to the theory of toric varieties Annals of Math. Studies Springer-Verlag., 131.Google Scholar
  10. 10.
    von zur Gathen, J., Gerhard, J. (1999): Modern Computer Algebra. Cambridge University PressGoogle Scholar
  11. 11.
    Gelfand, M. Kapranov, M. M., Zelevinsky, A. V. (1994): Discriminants, Resultants and multidimensional determinants. BirkhauserGoogle Scholar
  12. 12.
    Aroca Bisquert, F. (2000): Metodos algebraicos en ecuaciones differenciales ordinaries en el campo complejo. Thesis, Universidad de ValladolidGoogle Scholar
  13. 13.
    Cano, J. (1992): An extension of the Newton-Puiseux polygon construction to gives solutions of Pfaffian forms. Preprint, Universidad de ValladolidGoogle Scholar
  14. 14.
    Geddes, K. O. Czapor, S. R. Labahn, G. (1992): Algorithms for computer algebra. Kluwer Academic PublishersGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Frédéric Beringer
    • 1
  • Françoise Richard-Jung
    • 1
  1. 1.LMC-IMAGGrenoble cedexFrance

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