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 $f_{c}(x)=x^{2}+c$ 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.


Introduction
The dynamics of quadratic polynomials is commonly studied by using the family of maps ( ) = 2 + , where ∈ C and +1 = ( ) = 2 + . In the article [1] we presented the corresponding iterating system on a new coordinate plane using the change of variables to the ( , )-plane model (see [2]). In this new ( , V)-plane model, equations of periodic curves are of remarkably lower degree than in earlier models. Now the dynamics of the ( , V)plane is determined by the iteration of the function which is a two-dimensional quadratic polynomial map defined in the complex 2-space C 2 . The new iteration system is defined recursively as follows: where and ∈ N ∪ {0}. Now ( , V) is fixed , so ( , V) = ( , V), if and only if ( ( , V), ( , V)) = ( , V). 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][3][4][5][6]). In the article [5] Giarrusso and Fisher used it for the parameterization of the period 3 hyperbolic components of the Mandelbrot set. Later, in the article [2], Erkama studied the case of the period 3-4 hyperbolic components of the Mandelbrot set on the ( , )-plane and completely solved both cases.
Moreover, Erkama [2] has shown that the sum of periodic orbit points Journal of Complex Analysis is unique when = 3 or = 4. 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 periodfive 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 [7]. 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.

A Brief Introduction to the Elimination Theorem
We start with the Hilbert basis theorem: Every ideal ⊂ C[ 1 , . . . , ] has a finite generating set. That is, = ⟨ 1 , . . . , ⟩ for some 1 , . . . , ∈ . Hence ⟨ 1 , . . . , ⟩ is the ideal generated by the elements 1 , . . . , ; in other words 1 , . . . , is the basis of the ideal. The so called Gröbnerbasis has proved to be especially useful in many connections [7], for example, in kinematic analysis of mechanisms (see [8,9]). In order to introduce this basis we need the following constructions.
Let ∈ C[ 1 , . . . , ] be the polynomial given by where ∈ C, = ( 1 , . . . , ), and = 1 1 ⋅ 2 2 ⋅ ⋅ ⋅ is a monomial. Then the multidegree of is the leading coefficient of is the leading monomial of is and 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 Gröbnerbasis 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 N is a relation ≺ if (1) ≺ is linear ordering, 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 C[ 1 , . . . , , 1 , . . . , ]. Now we can define the product ordering as follows: There are several monomial orders but we need only the lexicographic order ≺ lex in the elimination theory. Let , ∈ N . Then we say that ⪰ and +1 ≺ +1 . One of the most important tools in the elimination theory is the Gröbner-basis of an ideal: Fix a monomial order. A finite subset of an ideal is said to be a Gröbner-basis (or standard basis) if Based on the Hilbert basis theorem we know that every ideal ⊂ C[ 1 , . . . , ] has a Gröbner-basis = { 1 , . . . , } so that It is essential to construct also an affine variety corresponding to the ideal. Let 1 , . . . , be polynomials in the ring C[ 1 , . . . , ]. Then we set Next we give an important elimination theorem which we use in our proof.
is 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 ( ).

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 [2] and the ( , V)-plane [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 [2] as Based on article [1], the equations of periodic orbits of period three and four are 3 ( , V) = 0 and 4 ( , V) = 0, where Now we form polynomials 3 ( , V, 3 ) = 0 and 4 ( , V, 4 ) = 0 based on formulas (25) as and obtain the ideals We eliminate from these ideals the variable and obtain the Gröbner-basis of the eliminated ideals 3 and 4 to calculate the Gröbner-basis of the ideals 3 and 4 using the Singular program ( [10]). Gröbner-bases of the ideals 3 and 4 , by using the ordering ≺ lex , where 3 ≺ lex V ≺ lex and 4 ≺ lex V ≺ lex , are where where Thus 31 and 41 depend only on the variables V and 5 . Based on the elimination Theorem 1 the set is the Gröbner-basis of the elimination ideal 4 and so ( 4 ) = ( 41 ). In the case 31 = 0 it follows that If 41 = 0 we have As 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 3 and 4 uniquely determine the orbit. If we eliminate in the first case the variable V instead of the variable , we obtain the Gröbner-basis which gives the same result as (36). However, the same procedure in the period four case produces the Gröbner-basis and this is of higher degree than (37).

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 [1], the equation for period-five orbit on the ( , V)-plane is of the form 5 ( , V) = 0, where Journal of Complex Analysis 5 According to the Theorem 3, the sum of the period-five points satisfies and based on the formula (3) we obtain on the ( , V)-plane. We form from this the polynomial Now we can form the pair of equations and the two polynomials 5 ( , V) and 5 ( , V, 5 ) form an ideal where 0 = V 7 + 2V 6 + V 5 We eliminate from this the variable by forming the Gröbner-basis 5 of the elimination ideal 5 in order to calculate the Gröbner-basis 5 of the ideal 5 using Singular program. We obtain the Gröbner-basis of the ideal as using ordering ≺ lex , where 5 ≺ lex V ≺ lex . Here 51 , 52 , 53 , 54 , and 55 depend on the variables , V, and 5 , and 56 depends only on the variables V and 5 . By the elimination theorem the set is the Gröbner-basis of the elimination ideal 5 and so ( 5 ) = ( 56 ). Now the Gröbner-basis of the elimination ideal 5 is of the form By (50) 5 is formed as a product of three terms. We denote the last of these terms in (50) by (V, 5 ). Now we obtain the variety ( ) of the elimination ideal as the union of three varieties corresponding to the factors of 5 as follows: Note that 5 is of degree 23 with respect to the variable V and of degree 7 with respect to the variable 5 . We denote, according to the extension theorem, = (V, 5 ) + terms such that deg ( ) < , where The corresponding varieties are In other words for all V ̸ = 0 and V ̸ = 1 we have (V, 5 ) ∉ V( 1 , 2 ) and in that case by the extension theorem then there exists ∈ C so that ( , V, 5 ) ∈ V( 5 ), so all partial solutions V( 5 ) = ((V, 5 ) | V ̸ = 0, V ̸ = 1) extend as solutions of the original system (45). Since the term (V, 5 ) is of degree 15 with respect to the variable V, it follows by the fundamental theorem of algebra that the equation (V, 5 ) = 0 has at most 15 different roots. For example, for the value 5 = 0 we obtain the Gröbner-basis of the elimination polynomial 5 = 27V 15 − 684V 13 + 556V 12 + 4002V 11 − 4336V 10 − 8380V 9 + 14868V 8 + 4003V 7 − 19924V 6 for which the variety V( 5 ) includes 15 different values. From these five are real and the rest ten are complex numbers. According to the extension theorem, for every pair of points (V 1 , 0), . . . , (V 15 , 0) we find the corresponding value of the variable so that ( 1 , V 1 , 0), . . . , ( 15 , V 15 , 0) ∈ V( 5 ). 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 V from the pair of equations (45) using the ordering ≺ lex , where 5 ≺ lex ≺ lex V.

Conflicts of Interest
The author declares that there are no conflicts of interest regarding the publication of this paper.