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On the Number of Zeros of an Analytic Perturbation of the Identically Zero Function on a Compact Set

A. Yu. Fishkin

M. V. Lomonosov Moscow State University

Abstract: An upper bound for the number of isolated zeros of an analytic perturbation $f(z,t)$ of the function $f(z,0)\equiv0$ on a compact set $\{z\in K\Subset\mathbb C\}$ is obtained for small values of the parameter $t\in\mathbb C^n$. The bound depends on an information about the Bautin ideal for the Taylor expansion of the function $f$ with respect to $z$ at one point of the compact set $K$ (e.g., at $0$) and on the maximal absolute value of $f$ in a neighborhood of $K$.

Keywords: analytic perturbation, holomorphic function, Bautin ideal, Dulac ideal, polydisk, germ of an analytic function, Noetherian ring, maximum principle.

UDC: 517.550.6

Received: 01.06.2007
Revised: 20.05.2008

DOI: 10.4213/mzm6582

 English version: Mathematical Notes, 2009, 85:1, 101–108

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