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) ≡ 0 on a compact set {z ∈ K ? ?} is obtained for small values of the parameter t ∈ ? 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.