Uniqueness of the Sum of Points of the Period-Five Cycle of Quadratic Polynomials

It is well known that the sum of points of the period-five cycle of the quadratic polynomial fc(x)=x2+c is generally not one-valued. In this paper we will show that the sum of cycle points of the curves of period five is at most three-valued on a new coordinate plane and that this result is essentially the best possible. The method of our proof relies on a implementing Gröbner-bases and especially extension theory from the theory of polynomial algebra.