Many of ordinary differential equations one can reduce to the form of a polynomial differential system. One of the best methods for the numerical solution of such systems is the method of Taylor series. In this work we consider the Cauchy problem for the polynomial ODE system, and then - a theorem about the accuracy of its solutions by this method. In connection with this theorem, a special minimax problem is formulated. Using this problem one can improve significantly the estimates of the theorem. The solutions of the corresponding minimax problems for a number of ODE systems are given.