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

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 $K$.