Uniqueness of the sum of points of the period five cycle of quadratic polynomials

TL;DR

This paper investigates the sum of points in the period five cycle of quadratic polynomials, showing it is at most three-valued on a new coordinate plane using algebraic methods, which is nearly optimal.

Contribution

It introduces a novel coordinate transformation and algebraic approach to analyze the sum of cycle points, establishing bounds and optimality for period five cycles.

Findings

Sum of cycle points is at most three-valued in the new coordinate plane

The result is essentially the best possible bound

Uses Gr"obner-bases and polynomial algebra extension theory

Abstract

It is well known that the sum of points of the period five cycle of the quadratic polynomial $f_{c}(x)=x^{2}+c$ 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 Gr\"obner-bases and especially extension theory from the theory of polynomial algebra.