Research Article | Open Access
Pekka Kosunen, "Uniqueness of the Sum of Points of the Period-Five Cycle of Quadratic Polynomials", Journal of Complex Analysis, vol. 2017, Article ID 8474868, 7 pages, 2017. https://doi.org/10.1155/2017/8474868
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 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.
The dynamics of quadratic polynomials is commonly studied by using the family of maps , where and . In the article  we presented the corresponding iterating system on a new coordinate plane using the change of variablesto the -plane model (see ). In this new -plane model, equations of periodic curves are of remarkably lower degree than in earlier models. Now the dynamics of the -plane is determined by the iteration of the functionwhich is a two-dimensional quadratic polynomial map defined in the complex -space . The new iteration system is defined recursively as follows:whereand . Now is fixed , so , if and only if . The set of such points is the union of all orbits, whose period divides , and the set of periodic points of period are the points with exact period dividing .
In complex dynamics, the sum of period cycle points has been a commonly used parameter in many connections (see, e.g., [2–6]). In the article  Giarrusso and Fisher used it for the parameterization of the period hyperbolic components of the Mandelbrot set. Later, in the article , Erkama studied the case of the period hyperbolic components of the Mandelbrot set on the -plane and completely solved both cases.
Moreover, Erkama  has shown that the sum of periodic orbit pointsis unique when or . Conversely, the sum of cyclic points of periods three and four determines these orbits uniquely. In the period-five case this situation changes and the sum of the cycle points is no longer unique. We can see this property in the articles [3, 6], in which Brown and Morton have formed the so called trace formulas in the cases of periods five and six using and the sum of period cycle points as parameters. In this paper we will show that, by implementing the change of variables (1), we obtain a new coordinate plane where the sum of period-five cycle points is at most three-valued and show that no better result is obtainable in this coordinate plane. This is done by applying methods of polynomial algebra (without the classical trace formula), as our proof relies on the use of the elimination theory and especially the extension theorem . The extension theorem tells us the best possible result (which the trace formula does not necessarily do) due to the use of Gröbner-basis. In the next section we present the most central tools and constructions related to these theorems.
2. A Brief Introduction to the Elimination Theorem
We start with the Hilbert basis theorem: Every ideal has a finite generating set. That is, for some . Hence is the ideal generated by the elements ; in other words is the basis of the ideal. The so called Gröbner-basis has proved to be especially useful in many connections , for example, in kinematic analysis of mechanisms (see [8, 9]). In order to introduce this basis we need the following constructions.
Let be the polynomial given bywhere , , and is a monomial. Then the multidegree of isthe leading coefficient of isthe leading monomial of isand the leading term of is
To calculate a Gröbner-basis of an ideal we need to order terms of polynomials by using a monomial ordering. A Gröbner-basis can be calculated by using any monomial ordering, but differences in the number of operations can be very significant. An effective tool to calculate the Gröbner-basis is the software Singular, which has been especially designed for operating with polynomial equations. Next we will define a monomial ordering of nonlinear polynomials.
Relation is the linear ordering in the set , if , , or for all . A monomial ordering in the set is a relation if(1) is linear ordering,(2)implication holds for all ,(3).
To compute elimination ideals we need product orderings. Let be an ordering for the variable , and let be ordering for the variable in the ring . Now we can define the product ordering as follows:There are several monomial orders but we need only the lexicographic order in the elimination theory. Let . Then we say that if and . One of the most important tools in the elimination theory is the Gröbner-basis of an ideal: Fix a monomial order. A finite subsetof an ideal is said to be a Gröbner-basis (or standard basis) ifBased on the Hilbert basis theorem we know that every ideal has a Gröbner-basis so that
It is essential to construct also an affine variety corresponding to the ideal. Let be polynomials in the ring . Then we setand we call as the affine variety defined by . Now if , and naturally we obtain the variety of the ideal as the variety of its Gröbner-basis: .
When we consider ideals and their algebraic varieties we are sometimes just interested about polynomials , which belong to the original ideal but contain only certain variables of the ring variables of . For this purpose we need elimination ideals. Let . The :th elimination ideal is the ideal of defined byNext we give an important elimination theorem which we use in our proof.
Theorem 1 (the elimination theorem). Let be an ideal and let be a Gröbner-basis of with respect to lexicographic order, where . Then, for every , the setis a Gröbner-basis of the :th elimination ideal .
The elimination theorem is closely related to the extension theorem, which tells us the correspondence between varieties of the original ideal and the elimination ideal. In other words, if we apply this theorem to a system of equations we see whether the partial solution of the system of equations is also a solution of the whole system .
Theorem 2 (the extension theorem). Let and let be the first elimination ideal of . For each write in the formwhere and , . Suppose that we have a partial solution . If , then there exists such that .
3. On Properties of Points Sums of Periods 3–5 Cycles
In this section we first prove the uniqueness properties of points sums of cycles of period three and four by using methods from polynomial algebra in a new way. After this we concentrate on the period-five case and show that the sum of period-five cycle points is at most three-valued. The next result shows the relation between the sums of cycle points of the -plane  and the -plane .
Proof. By writing out both components we obtainand similarly
3.1. The Uniqueness of Cycle Points Sums of Periods Three and Four Orbits
The sums of points of the periods three and four cycles is obtained in  as
According to Theorem 3 and by using the formula (3) we obtain on the -planeBased on article , the equations of periodic orbits of period three and four are and , whereNow we form polynomials and based on formulas (25) asBased on the previous equations we can form the pair of equationsand obtain the idealsWe eliminate from these ideals the variable and obtain the Gröbner-basis of the eliminated ideals and to calculate the Gröbner-basis of the ideals and using the Singular program (). Gröbner-bases of the ideals and , by using the ordering , where and , arewhereandwhereThus and depend only on the variables and . Based on the elimination Theorem 1 the setis the Gröbner-basis of the elimination ideal and so . At the same way the setis the Gröbner-basis of the elimination ideal and so . In the case it follows thatIf we haveAs we can see, in both cases the sum of the points of cycles of the given period is unique. In other words, the orbit sums and uniquely determine the orbit. If we eliminate in the first case the variable instead of the variable , we obtain the Gröbner-basiswhich gives the same result as (36). However, the same procedure in the period four case produces the Gröbner-basisand this is of higher degree than (37).
3.2. On the Uniqueness of the Cycle Points Sum of Period-Five Orbits
Next we prove that, in the case of period-five cycles, the sum of period-five points is at most three-valued. We use in this proof the Gröbner-basis of an ideal, like before in periods three and four cases, which produce for us the Gröbner-basis of the elimination ideal. Because this method relies on bases, the following result is optimal.
Theorem 4. The sum of period-five cycle points is at most three-valued.
Proof. By article , the equation for period-five orbit on the -plane is of the form , whereAccording to the Theorem 3, the sumof the period-five points satisfies and based on the formula (3) we obtainon the -plane. We form from this the polynomialNow we can form the pair of equationsand the two polynomials and form an idealwhereWe eliminate from this the variable by forming the Gröbner-basis of the elimination ideal in order to calculate the Gröbner-basis of the ideal using Singular program. We obtain the Gröbner-basis of the ideal asusing ordering , where . Here , , , , and depend on the variables , , and , and depends only on the variables and . By the elimination theorem the setis the Gröbner-basis of the elimination ideal and so . Now the Gröbner-basis of the elimination ideal is of the formwhereBy (50) is formed as a product of three terms. We denote the last of these terms in (50) by . Now we obtain the variety of the elimination ideal as the union of three varieties corresponding to the factors of as follows:Note that is of degree with respect to the variable and of degree with respect to the variable . We denote, according to the extension theorem,whereThe corresponding varieties aresoIn other words for all and we have and in that case by the extension theorem then there exists so that , so all partial solutions extend as solutions of the original system (45). Since the term is of degree with respect to the variable , it follows by the fundamental theorem of algebra that the equation has at most different roots. For example, for the value we obtain the Gröbner-basis of the elimination polynomial for which the variety includes different values. From these five are real and the rest ten are complex numbers. According to the extension theorem, for every pair of points we find the corresponding value of the variable so that . Consequently the sum of period-five cycle points attains the same value at most three times.
We obtain also the same result if we eliminate the variable from the pair of equations (45) using the ordering , where .
Conflicts of Interest
The author declares that there are no conflicts of interest regarding the publication of this paper.
- P. Kosunen, “Periodic orbits 1-5 of quadratic polynomials on a new coordinate plane,” https://arxiv.org/abs/1703.05146.
- T. Erkama, “Periodic orbits of quadratic polynomials,” Bulletin of the London Mathematical Society, vol. 38, no. 5, pp. 804–814, 2006.
- A. Brown, “Equations for periodic solutions of a logistic difference equation,” The Journal of the Australian Mathematical Society, Series B — Applied Mathematics, vol. 23, no. 1, pp. 78–94, 1981/82.
- E. V. Flynn, B. Poonen, and E. F. Schaefer, “Cycles of quadratic polynomials and rational points on a genus-2 curve,” Duke Mathematical Journal, vol. 90, no. 3, pp. 435–463, 1997.
- D. Giarrusso and Y. Fisher, “A parameterization of the period 3 hyperbolic components of the mandelbrot set,” Proceedings of the American Mathematical Society, vol. 123, no. 12, pp. 3731–3737, 1995.
- P. Morton, “Arithmetic properties of periodic points of quadratic maps,” Acta Arithmetica, vol. 87, no. 2, pp. 89–102, 1998.
- D. Cox, J. Little, and D. O'Shea, Ideals, varieties and algorithms, 1997.
- T. Arponen, A. Müller, S. Piipponen, and J. Tuomela, “Computational algebraic geometry and global analysis of regional manipulators,” Applied Mathematics and Computation, vol. 232, pp. 820–835, 2014.
- T. Arponen, A. Müller, S. Piipponen, and J. Tuomela, “Kinematical analysis of overconstrained and underconstrained mechanisms by means of computational algebraic geometry,” Meccanica, vol. 49, no. 4, pp. 843–862, 2014.
- W. Decker, G.-M. Greuel, G. Pfister, and H. Schönemann, Singular 4-0-2—A computer algebra system for polynomial computations, 2015, http://www.singular.uni-kl.de.
Copyright © 2017 Pekka Kosunen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.