The paper deals with the complexity of homotopy methods for solving systems of polynomial equations. For general homotopy equations of the form \(f_ t (\zeta) = 0\), \(t \in [0,1]\), whose zeros are denoted by \(\zeta_ t\), the number \(l\) of steps (with \(0 = t_ 0 < t_ 1 < \cdots < t_ l = 1)\), that are necessary such that an approximate of the zero in the previous step is an appropriate starting point for Newton’s method in the next step, is estimated in terms of the distance of the curve \((f_ t, \zeta_ t)\) to the variety of ill-posed problems, the length of the curve \((f_ t)\), the maximum degree of the polynomials \(f_ t\) and a numerically known constant. The proof is based on a Kantorovich- type local convergence theorem for analytic functions.