On a rational mapping of a polynomial system into a quadratic system

In this paper we prove that a polynomial system can be always mapped by some rational mapping into a general quadratic system. This result is useful for studies of systems with a complex dynamics. One generalization is discussed. Our approach is based on theorems on algebraic dependent polynomials. Key-Words:Polynomial, polynomial system, quadratic system, rational mapping, algebraic dependence, complex dynamics Theorem 1. (Perron, [6]; see the modern version in [7]). Let F1(x1,...,xn),...,Fn(x1,...,xn),Fn+1(x1,...,xn)∈ R[X] be polynomials of positive degrees of variables (x1,...,xn ) Let v be the weight of the ring R[W] defined by conditions v(wi) = deg Fi for i =1, ..., n + 1. Then there is a nontrivial polynomial T ∈ R[W], W = (w1, ...,wn+1) such that T(F1(x1, ..., xn), ..., Fn+1(x1, ..., xn)) ≡ 0,