manuscripta mathematica

, Volume 32, Issue 1–2, pp 91–100

Puiseux expansion for space curves

  • Joseph Maurer
Article

Abstract

For any ideal I in a convergent power series ring ℌ {X1,..,Xn} (n≥2) with one dimensional zero set X ⊂ (ℌn, 0) we give a method of computing a parametrization of each irreducible component of the reduction of X. This generalizes the well-known method of the Newton polygon or the so called Puiseux expansion for plane curves (see [N], [P], and [B]). The slope of a side of the Newton polygon is generalized to what we calltropism of the ideal. It may be visualized as the direction of a hyperplane touching the Newton polyhedron of every element of the ideal at least along an edge.

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References

  1. [B] BRIESKORN, E.: Al gebraische Kurven. Vorlesungsausarbeitung Bonn WS 1975/76, SS 1976; Math. Inst. Univ. Bonn 1978/79Google Scholar
  2. [M] MACMILLAN, W. D.: A method for determining the solutions of a system of analytic functions in the neighborhood of a branch point. Math. Ann.72 (1912) 180–202Google Scholar
  3. [N] NEWTON, I.: The correspondence of Isaak Newton, Vol. 2 (1676–1687), Cambridge University Press 1960, pp. 20–42 and 110–163Google Scholar
  4. [P] PUISEUX, V.: Recherches sur les fonctions algébriques. Journal de math. (1) 15 (1850) 365–480Google Scholar
  5. [S] SATHER, D.: Branching of solutions of nonlinear equations. Rocky Mountain Journal of Mathematics3 (1973) 203–250Google Scholar

Copyright information

© Springer-Verlag 1980

Authors and Affiliations

  • Joseph Maurer
    • 1
  1. 1.Mathematisches InstitutUniversität DüsseldorfDüsseldorfFederal Republic of Germany

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