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 Mat. Zametki, 2001, Volume 69, Issue 2, Pages 194–199 (Mi mz495)

On the Weierstrass Preparation Theorem

A. A. Mailybaev, S. S. Grigoryan

Research Institute of Mechanics, M. V. Lomonosov Moscow State University

Abstract: 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.

DOI: https://doi.org/10.4213/mzm495

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English version:
Mathematical Notes, 2001, 69:2, 170–174

Bibliographic databases:

UDC: 517.55

Citation: A. A. Mailybaev, S. S. Grigoryan, “On the Weierstrass Preparation Theorem”, Mat. Zametki, 69:2 (2001), 194–199; Math. Notes, 69:2 (2001), 170–174

Citation in format AMSBIB
\Bibitem{MaiGri01} \by A.~A.~Mailybaev, S.~S.~Grigoryan \paper On the Weierstrass Preparation Theorem \jour Mat. Zametki \yr 2001 \vol 69 \issue 2 \pages 194--199 \mathnet{http://mi.mathnet.ru/mz495} \crossref{https://doi.org/10.4213/mzm495} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1830219} \zmath{https://zbmath.org/?q=an:0999.32004} \transl \jour Math. Notes \yr 2001 \vol 69 \issue 2 \pages 170--174 \crossref{https://doi.org/10.1023/A:1002816101132} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000168619500018} 

• http://mi.mathnet.ru/eng/mz495
• https://doi.org/10.4213/mzm495
• http://mi.mathnet.ru/eng/mz/v69/i2/p194

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This publication is cited in the following articles:
1. Cai T., Zhang H., Wang B., Yang F., “The Asymptotic Analysis of Multiple Imaginary Characteristic Roots for Lti Delayed Systems Based on Puiseux-Newton Diagram”, Int. J. Syst. Sci., 45:5 (2014), 1145–1155
2. Martinez-Gonzalez A., Mendez-Barrios C.F., Niculescu S.I., Chen J., “Some Remarks on the Asymptotic Behavior For Quasipolynomials With Two Delays”, IFAC PAPERSONLINE, 51:14 (2018), 318–323
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