Geodesic
On the Number of Zeros of an Analytic Perturbation of the Identically Zero Function on a Compact Set
Matematičeskie zametki, Tome 85 (2009) no. 1, pp. 110-118

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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 $K$.
Mots-clés : analytic perturbation
Keywords: holomorphic function, Bautin ideal, Dulac ideal, polydisk, germ of an analytic function, Noetherian ring, maximum principle. 
A. Yu. Fishkin. On the Number of Zeros of an Analytic Perturbation of the Identically Zero Function on a Compact Set. Matematičeskie zametki, Tome 85 (2009) no. 1, pp. 110-118. http://geodesic.mathdoc.fr/item/MZM_2009_85_1_a9/
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