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. 
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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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