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

1. Introduction

The dynamics of quadratic polynomials is commonly studied by using the family of maps [f.sub.c](x) = [x.sup.2] + c, where c [member of] C and [x.sub.i+1] = [f.sub.c]([x.sub.i]) = [x.sup.2.sub.i] + c. In the article [1] we presented the corresponding iterating system on a new coordinate plane using the change of variables

u = x + y = [x.sub.0] + [x.sub.1]

v = x + [y.sup.2] + y - [x.sup.2] = [x.sub.0] + [x.sup.2.sub.1] + [x.sub.1] - [x.sup.2.sub.0] (1)

to the (x, y)-plane model (see [2]). In this new (u, v)-plane model, equations of periodic curves are of remarkably lower degree than in earlier models. Now the dynamics of the (u, v)-plane is determined by the iteration of the function

G(u, v) = (R (u, v), Q (u, v))

= ([-u + v + uv/u], [[u.sup.2] - u + v - [u.sup.2]v - uv + u[v.sup.2] + [v.sup.2]/u]), (2)

which is a two-dimensional quadratic polynomial map defined in the complex 2-space [C.sup.2]. The new iteration system is defined recursively as follows:

[mathematical expression not reproducible] (3)

where

[R.sub.n+1](u, v) = [Q.sub.n](u, v) - 1 + [[Q.sub.n](u, v)/[R.sub.n](u, v)]

[Q.sub.n+1](u, v) = [R.sub.n+1](u, v)(1 + [Q.sub.n](u, v) - [R.sub.n](u, v)), (4)

and n [member of] N [union] {0}. Now (u, v) is fixed [G.sup.n], so [G.sup.n](u, v) = (u, v), if and only if ([R.sub.n](u, v), [Q.sub.n](u, v)) = (u, v). The set of such points is the union of all orbits, whose period divides n, and the set of periodic points of period n are the points with exact period dividing n.

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 [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 (x, y)-plane and completely solved both cases.

Moreover, Erkama [2] has shown that the sum of periodic orbit points

[S.sub.n] = [x.sub.0] + [x.sub.1] + [x.sub.2] + ... + [x.sub.n-2] + [x.sub.n-1] (5)

is unique when n = 3 or n = 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 c 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 [7]. The extension theorem tells us the best possible result (which the trace formula does not necessarily do) due to the use of Grobner-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 I [subset] C[[x.sub.1], ..., [x.sub.n]] has a finite generating set. That is, I = <[g.sub.1], ..., [g.sub.t]> for some [g.sub.1], ..., [g.sub.t] [member of] I. Hence <[g.sub.1], ..., [g.sub.t]> is the ideal generated by the elements [g.sub.1], ..., [g.sub.t]; in other words [g.sub.1], ..., [g.sub.t] is the basis of the ideal. The so called Grobner-basis 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 f [member of] C[[x.sub.1], ..., [x.sub.n]] be the polynomial given by

f = [summation over ([alpha])][a.sub.[alpha]][x.sup.[alpha]], (6)

where [a.sub.[alpha]] [member of] C, [alpha] = ([[alpha].sub.1], ..., [[alpha].sub.n)], and [mathematical expression not reproducible] is a monomial. Then the multidegree of f is

mdeg(f) = max {[alpha] | [a.sub.[alpha]] [not equal to] 0}, (7)

the leading coefficient of f is

LC(f) = [a.sub.mdeg(f)], (8)

the leading monomial of f is

LM(f) = [x.sup.mdes(f)], (9)

and the leading term of f is

LT(f) = LC(f) LM(f). (10)

To calculate a Grobner-basis of an ideal we need to order terms of polynomials by using a monomial ordering. A Grobner-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 Groobner-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 S, if x < y, x = y, or y < x for all x, y [member of] S. A monomial ordering in the set [N.sup.n] is a relation [??] if

(1) [??] is linear ordering,

(2) implication [mathematical expression not reproducible] holds for all [alpha], [beta], [gamma] [member of] [N.sup.n],

(3) [x.sup.[alpha]] > 1.

To compute elimination ideals we need product orderings. Let [[??].sub.A] be an ordering for the variable x, and let [[??].sub.B] be ordering for the variable y in the ring C[[x.sub.1], ..., [x.sub.n], [y.sub.1], ..., [y.sub.m]]. Now we can define the product ordering as follows:

[mathematical expression not reproducible] (11)

There are several monomial orders but we need only the lexicographic order [[??].sub.lex] in the elimination theory. Let [alpha], [beta] [member of] [N.sup.n]. Then we say that [x.sup.[alpha]] [[??].sub.lex] [x.sup.[beta]] if [mathematical expression not reproducible]. One of the most important tools in the elimination theory is the Grobner-basis of an ideal: Fix a monomial order. A finite subset

[G.sub.I] = {[g.sub.1], ..., [g.sub.t]} [subset] I (12)

of an ideal I is said to be a Grobner-basis (or standard basis) if

<LT([g.sub.1]), ..., LT([g.sub.t])> = <LT (I)>. (13)

Based on the Hilbert basis theorem we know that every ideal I [subset] C[[x.sub.1], ..., [x.sub.n]] has a Grobner-basis [G.sub.I] = {[g.sub.1], ..., [g.sub.s]} so that

<[G.sub.I]> = I. (14)

It is essential to construct also an affine variety corresponding to the ideal. Let [f.sub.1], ..., [f.sub.s] be polynomials in the ring C[[x.sub.1], ..., [x.sub.n]]. Then we set

V([f.sub.1], ..., [f.sub.s])

= {([a.sub.1], ..., [a.sub.n]) [member of] [C.sup.n] : [f.sub.i]([a.sub.1], ..., [a.sub.n]) = 0 [for all]1 [less than or equal to] i [less than or equal to] s}, (15)

and we call V([f.sub.1], ..., [f.sub.s]) as the affine variety defined by [f.sub.1], ..., [f.sub.s]. Now if I = <[f.sub.1], ..., [f.sub.s]>, V(I) = V([f.sub.1], ..., [f.sub.s]) and naturally we obtain the variety of the ideal as the variety of its Grobner-basis: V(I) = V(<[G.sub.I]>).

When we consider ideals and their algebraic varieties we are sometimes just interested about polynomials f [member of] C[[x.sub.1], ..., [x.sub.n]], which belong to the original ideal f [member of] I but contain only certain variables of the ring variables of C[[x.sub.1], ..., [x.sub.n]]. For this purpose we need elimination ideals. Let I = <[f.sub.1], ..., [f.sub.s]> [subset] C[[x.sub.1], ..., [x.sub.n]]. The k:th elimination ideal [I.sub.k] is the ideal of C[[x.sub.k+1], ..., [x.sub.n]] defined by

[I.sub.k] = I [intersection] C [[x.sub.k+1], ..., [x.sub.n]]. (16)

Next we give an important elimination theorem which we use in our proof.

Theorem 1 (the elimination theorem). Let I [subset] C[[x.sub.1], ..., [x.sub.n]] be an ideal and let G be a Grobner-basis of I with respect to lexicographic order, where [mathematical expression not reproducible]. Then, for every 0 [less than or equal to] k [less than or equal to] n, the set

[G.sub.k] = G [intersection] C [[x.sub.k+1], ..., [x.sub.n]] (17)

is a Grobner-basis of the k:th elimination ideal [I.sub.k].

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 V([I.sub.k]) of the system of equations is also a solution of the whole system V(I).

Theorem 2 (the extension theorem). Let I = <[f.sub.1], ..., [f.sub.s]> [subset] C[[x.sub.1], ..., [x.sub.n]] and let [I.sub.1] be the first elimination ideal of I. For each 1 [less than or equal to] i [less than or equal to] s write [f.sub.i] in the form

[mathematical expression not reproducible] (18)

where [N.sub.i] [greater than or equal to] 0 and [g.sub.i] [xi] C[[x.sub.2], ..., [x.sub.n]], [g.sub.i] [not equal to] 0. Suppose that we have a partial solution ([a.sub.2], ..., [a.sub.n]) [member of] V([I.sub.1]). If ([a.sub.2], ..., [a.sub.n]) [not member of] V([g.sub.1], ..., [g.sub.s]), then there exists [a.sub.1] [member of] C such that ([a.sub.1], [a.sub.2], ..., [a.sub.n]) [member of] V(I).

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 (x, y)-plane [2] and the (u, v)-plane [1].

Theorem 3. Let [x.sub.0], [x.sub.1], [x.sub.2], ..., [x.sub.n-1] be the period-n orbit points. If

[S.sub.n] = [x.sub.0] + [x.sub.1] + [x.sub.2] + ... + [x.sub.n-2] + [x.sub.n-1], (19)

then by transformation of (1) and (3)

[S.sub.n] = [1/2][S.sup.1.sub.n] = [1/2][S.sup.2.sub.n], (20)

where

Proof. By writing out both components we obtain

[S.sup.1.sub.n] = [u.sub.0] + [u.sub.1] + [u.sub.2] + ... + [u.sub.n-2] + [u.sub.n-1],

[S.sup.2.sub.n] = [v.sub.0] + [v.sub.1] + [v.sub.2] + ... + [v.sub.n-2] + [v.sub.n-1]. (21)

Proof. By writing out both components we obtain

[mathematical expression not reproducible] (22)

and similarly

[mathematical expression not reproducible] (23)

3.1. The Uniqueness of CyclePoints Sums of Periods Three and Four Orbits. The sums of points of the periods three and four cycles is obtained in [2] as

[S.sub.3] = [x.sub.0] + [x.sub.1] + [x.sub.2],

[S.sub.4] = [x.sub.0] + [x.sub.1] + [x.sub.2] + [x.sub.3]. (24)

According to Theorem 3 and by using the formula (3) we obtain on the (u, v)-plane

[mathematical expression not reproducible] (25)

Based on article [1], the equations of periodic orbits of period three and four are [P.sub.3](u, v) = 0 and [P.sub.4](u, v) = 0, where

[mathematical expression not reproducible] (26)

Now we form polynomials [B.sub.3](u, v, [S.sub.3]) = 0 and [B.sub.4](u, v, [S.sub.4]) = 0 based on formulas (25) as

[mathematical expression not reproducible] (27)

Based on the previous equations we can form the pair of equations

[mathematical expression not reproducible] (28)

and obtain the ideals

[mathematical expression not reproducible] (29)

We eliminate from these ideals the variable u and obtain the Grobner-basis of the eliminated ideals [I.sub.3u] and [I.sub.4u] to calculate the Grobner-basis of the ideals [I.sub.3] and [I.sub.4] using the Singular program ([10]). Grobner-bases of the ideals [I.sub.3] and [I.sub.4], by using the ordering [[??].sub.lex], where [S.sub.3] [[??].sub.lex] v [[??].sub.lex] u and [S.sub.4] [[??].sub.lex] v [[??].sub.lex] u, are

[G.sub.3] = {[g.sub.31], [g.sub.32]}, (30)

where

[g.sub.31] = [v.sup.3] - 2[v.sup.2][S.sub.3] - 2v[S.sub.3] - 3v - 1,

[g.sub.32] = u + [v.sup.2] - 2v[S.sub.3] - 2[S.sub.3] - 2, (31)

and

[G.sub.4] = {[g.sub.41], [g.sub.42], [g.sub.43], [g.sub.44]}, (32)

where

[mathematical expression not reproducible] (33)

Thus [g.sub.31] and [g.sub.41] depend only on the variables v and [S.sub.5]. Based on the elimination Theorem 1 the set

[G.sub.3u] = [G.sub.3] [intersection] C [v, [S.sub.3]] = {[g.sub.31]} (34)

is the Grobner-basis of the elimination ideal [I.sub.3u] and so V([I.sub.3u]) = V([g.sub.31]). At the same way the set

[G.sub.4u] = [G.sub.4] [intersection] C [v, [S.sub.4]] = {[g.sub.41]} (35)

is the Grobner-basis of the elimination ideal [I.sub.4u] and so V([I.sub.4u]) = V([g.sub.41]). In the case [g.sub.31] = 0 it follows that

[S.sub.3] = -1 - 3v + [v.sup.3]/2v (v + 1). (36)

If [g.sub.41] = 0 we have

[S.sub.4] =[ -[v.sup.4] - [v.sup.3] + [v.sup.2] + v/-[v.sup.3] - [v.sup.2]] = [v.sup.2] - 1/v. (37)

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 [S.sub.3] and [S.sub.4] uniquely determine the orbit. If we eliminate in the first case the variable v instead of the variable u, we obtain the Groobner-basis

[G.sub.3v] = [u.sup.3] - 2[u.sup.2][S.sub.3] - 2u[S.sub.3] - 3u - 1, (38)

which gives the same result as (36). However, the same procedure in the period four case produces the Grobner-basis

[mathematical expression not reproducible] (39)

and 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 Grobner-basis of an ideal, like before in periods three and four cases, which produce for us the Grobner-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 (u, v)-plane is of the form [P.sub.5](u, v) = 0, where

[mathematical expression not reproducible] (40)

According to the Theorem 3, the sum

[S.sub.5] = [x.sub.0] + [x.sub.1] + [x.sub.2] + [x.sub.3] + [x.sub.4] (41)

of the period-five points satisfies

[S.sub.n] = [1/2][S.sup.1.sub.n] = [1/2][S.sup.2.sub.n], (42)

and based on the formula (3) we obtain

[mathematical expression not reproducible] (43)

on the (u, v)-plane. We form from this the polynomial

[mathematical expression not reproducible] (44)

Now we can form the pair of equations

[P.sub.5](u, v) = 0

[B.sub.5](u, v, [S.sub.5]) = 0, (45)

and the two polynomials [P.sub.5](u, v) and [B.sub.5](u, v, [S.sub.5]) form an ideal

[mathematical expression not reproducible] (46)

where

[mathematical expression not reproducible] (47)

We eliminate from this the variable u by forming the Grobner-basis [G.sub.5u] of the elimination ideal [I.sub.5u] in order to calculate the Grobner-basis [G.sub.5] of the ideal [I.sub.5] using Singular program. We obtain the Grobner-basis of the ideal I as

[G.sub.5] = {[g.sub.51], [g.sub.52], [g.sub.53], [g.sub.54], [g.sub.55], [g.sub.56]} (48)

using ordering [[??].sub.lex] where [S.sub.5] [[??].sub.lex] v [[??].sub.lex] u. Here [g.sub.51], [g.sub.52], [g.sub.53], [g.sub.54], and [g.sub.55] depend on the variables u, v, and [S.sub.5], and [g.sub.56] depends only on the variables v and [S.sub.5]. By the elimination theorem the set

[G.sub.5u] = G [intersection] C [v, [S.sub.5]] = {[g.sub.56]} (49)

is the Grobner-basis of the elimination ideal [I.sub.5u] and so V([I.sub.5u]) = V([g.sub.56]). Now the Grobner-basis of the elimination ideal [I.sub.5u] is of the form

[mathematical expression not reproducible] (50)

where

[mathematical expression not reproducible] (51)

By (50) [G.sub.5u] is formed as a product of three terms. We denote the last of these terms in (50) by C(v, [S.sub.5]). Now we obtain the variety V([I.sub.u]) of the elimination ideal as the union of three varieties corresponding to the factors of [G.sub.5u] as follows:

V([I.sub.5u]) = V([v.sup.6]) [union] v ([(v + 1).sup.2]) [union] V (C(v, [S.sub.5])) = {(0, [S.sub.5]), (-1, [S.sub.5])} [union] V (C(v, [S.sub.5])). (52)

Note that [G.sub.5u] is of degree 23 with respect to the variable v and of degree 7 with respect to the variable [S.sub.5]. We denote, according to the extension theorem,

[mathematical expression not reproducible] (53)

where

[g.sub.1] = [a.sub.7] = -[v.sup.4] + 2[v.sup.3] - [v.sup.2],

[g.sub.2] = [b.sub.4] = 2[v.sup.2] - 2v. (54)

The corresponding varieties are

V([g.sub.1]) = {(0, [S.sub.5]), (1, [S.sub.5])} = V([g.sub.2]), (55)

so

V([g.sub.1], [g.sub.2]) = V([g.sub.1]) [intersection] V([g.sub.2]) = {(0, [S.sub.5]), (1, [S.sub.5])}. (56)

In other words for all v [not equal to] 0 and v [not equal to] 1 we have (v, [S.sub.5]) [not member of] V([g.sub.1], [g.sub.2]) and in that case by the extension theorem then there exists u [member of] C so that (u, v, [S.sub.5]) [member of] V([I.sub.5]), so all partial solutions V([I.sub.5u]) = ((v, [S.sub.5]) | v [not equal to] 0, v [not equal to] 1) extend as solutions of the original system (45). Since the term C(v, [S.sub.5]) is of degree 15 with respect to the variable v, it follows by the fundamental theorem of algebra that the equation C(v, [S.sub.5]) = 0 has at most 15 different roots. For example, for the value [S.sub.5] = 0 we obtain the Grobner-basis of the elimination polynomial

[mathematical expression not reproducible] (57)

for which the variety V([I.sub.5u]) 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.sub.1], 0), ..., ([v.sub.15], 0) we find the corresponding value of the variable u so that ([u.sub.1], [v.sub.1], 0), ..., ([u.sub.15], [v.sub.15], 0) [member of] V([I.sub.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 [[??].sub.lex], where [S.sub.5] [[??].sub.hex] u [[??].sub.lex] v.

https://doi.org/10.1155/2017/8474868

Conflicts of Interest

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

References

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[2] T. Erkama, "Periodic orbits of quadratic polynomials," Bulletin of the London Mathematical Society, vol. 38, no. 5, pp. 804-814, 2006.

[3] 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.

[4] 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

[5] 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.

[6] P Morton, "Arithmetic properties of periodic points of quadratic maps," Acta Arithmetica, vol. 87, no. 2, pp. 89-102, 1998.

[7] D. Cox, J. Little, and D. O'Shea, Ideals, varieties and algorithms, 1997.

[8] T. Arponen, A. Muller, S. Piipponen, and J. Tuomela, "Computational algebraic geometry and global analysis of regional manipulators," Applied Mathematics and Computation, vol. 232, pp. 820-835, 2014.

[9] T. Arponen, A. Muller, 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.

[10] W. Decker, G.-M. Greuel, G. Pfister, and H. Schonemann, Singular 4-0-2--A computer algebra system for polynomial computations, 2015, http://www.singular.uni-kl.de.

Pekka Kosunen

Department of Physics and Mathematics, University of Eastern Finland, P.O. Box 111, 80101 Joensuu, Finland

Correspondence should be addressed to Pekka Kosunen; pekka.kosunen@uef.fi

Received 3 August 2017; Accepted 29 October 2017; Published 23 November 2017

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Publication:Journal of Complex Analysis
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Date:Jan 1, 2017
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