Estimates for the Orders of Zeros of Polynomials in Some Analytic Functions

In the present paper, we consider estimates for the orders of zeros of polynomials in functions satisfying a system of algebraic differential equations and possessing a special $D$-property defined in the paper. The main result obtained in the paper consists of two theorems for the two cases in which these estimates are given. These estimates are improved versions of a similar estimate proved earlier in the case of algebraically independent functions and a single point. They are derived from a more general theorem concerning the estimates of absolute values of ideals in the ring of polynomials, and the proof of this theorem occupies the main part of the present paper. The proof is based on the theory of ideals in rings of polynomials. Such estimates may be used to prove the algebraic independence of the values of functions at algebraic points.