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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     author = {A. Yu. Fishkin},
     title = {On the {Number} of {Zeros} of an {Analytic} {Perturbation} of the {Identically} {Zero} {Function} on a {Compact} {Set}},
     journal = {Matemati\v{c}eskie zametki},
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     volume = {85},
     number = {1},
     year = {2009},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2009_85_1_a9/}
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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/