Mathematical Notes

, Volume 69, Issue 1–2, pp 170–174 | Cite as

On the Weierstrass Preparation Theorem

  • S. S. Grigoryan
  • A. A. Mailybaev


An analytic function of several variables is considered. It is assumed that the function vanishes at some point. According to the Weierstrass preparation theorem, in the neighborhood of this point the function can be represented as a product of a nonvanishing analytic function and a polynomial in one of the variables. The coefficients of the polynomial are analytic functions of the remaining variables. In this paper we construct a method for finding the nonvanishing function and the coefficients of the polynomial in the form of Taylor series whose coefficients are found from an explicit recursive procedure using the derivatives of the initial function. As an application, an explicit formula describing a bifurcation diagram locally up to second-order terms is derived for the case of a double root.

Weierstrass preparation theorem analytic function of several variables bifurcation diagram 


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Copyright information

© Plenum Publishing Corporation 2001

Authors and Affiliations

  • S. S. Grigoryan
    • 1
  • A. A. Mailybaev
    • 1
  1. 1.Institute of MechanicsM. V. Lomonosov Moscow State UniversityRussia

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