Cyclicity and Maximal Multiplicity for Zeros of Families of Analytic Functions

Let f_{\lambda} be a family of holomorphic functions in the unit disk, holomorphic in parameter \lambda\in U\subset\C^{n}. We estimate the number of zeros of f_{\lambda} in a smaller disk via some characteristic of the ideal generated by Taylor coefficients of f_{\lambda}. Our estimate is locally sharp and improve the previous estimate obtained by Roytwarf and Yomdin.