# New Quasi-Coincidence Point Polynomial Problems

Authors: Yi-Chou Chen; Hang-Chin Lai
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

LetF:ℝ×ℝ→ℝ be a real-valued polynomial function of the form F(x,y)=as(x)ys+as-1(x)ys-1+⋯+a0(x), where the degree s of y in F(x,y) is greater than or equal to 1. For arbitrary polynomial function f(x)∈ℝ[x], x∈ℝ, we will find a polynomial solution y(x)∈ℝ[x] to satisfy the following equation: (*): F(x,y(x))=af(x), where a∈ℝ is a constant depending on the solution y(x), namely, a quasi-coincidence (point) solution of (*), and a is called a quasi-coincidence value. In this paper, we prove that (i) the leading coefficient as(x) must be a factor of f(x), and (ii) each solution of (*) is of the form y(x)=-as-1(x)/sas(x)+λp(x), where λ is arbitrary and p(x)=c(f(x)/as(x))1/s is also a factor of f(x), for some constant c∈ℝ, provided the equation (*) has infinitely many quasi-coincidence (point) solutions.


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## Body

## 1. Introduction and Preliminaries

LetF:ℚ(α)×ℚ(α)→ℚ(α) (where α is an algebraic number) be a polynomial function. Lenstra [1] investigated that F(x,y(x))=0. He solved a polynomial function y=y(x)∈ℚ(α)[x] and derived to find a polynomial y=y(x) satisfying an x∈ℚ(α)[x] as a fixed point of the polynomial equation. That is,
(1)F(x,y(x))=x
has a polynomial solution y(x)∈ℚ(α)[x].Further, Tung [2, 3] extended (1) to solve y(x)∈𝕂[x] (𝕂 is a field) for the following equation:
(2)F(x,y(x))=axm,
where a∈𝕂 is a constant depending on the polynomial solution y(x) and m∈ℕ a given positive integer.Recently, Lai and Chen [4, 5] extended (2) to solve y(x)∈ℝ[x] to satisfy the polynomial equation as the form:
(3)F(x,y(x))=apm(x),x∈ℝ,
where a∈ℝ, p(·) is an irreducible polynomial in x∈ℝ and the polynomial function F(x,y):ℝ×ℝ→ℝ is written by
(4)F(x,y)=∑i=0s‍ai(x)yiwiths≥1,
where s=degyF denotes the degree of y in F(x,y).Definition 1 (see [4]).
A polynomial functiony=y(x) satisfying (3) is called a quasi-fixed solution corresponding to some real number a. This number a is called a quasi-fixed value corresponding to the polynomial solutions y=y(x).In mathematics, a coincidence point (or simply coincidence) of two mappings is a point in their domain having the same image point under both mappings. Coincidence theory (the study of coincidence points) is, in most settings, a generalization of fixed point theory.In this paper, we define a more general coincidence (point) problem in which thef(x)∈ℝ[x] is replaced by the irreducible polynomial power pm(x)∈ℝ[x] throughout this paper, where f(x) is an arbitrary polynomial. Then, we restate (3) as the following equation:
(5)F(x,y)=af(x).
It is a new development coincidence point-like problem. We call the polynomial solution y=y(x) for (5) as a quasi-coincidence (point) solution. Precisely, we give the following definition like Definition 1.Definition 2.
A polynomial functiony=y(x) satisfying (5) is called a quasi-coincidence (point) solution corresponding to some real number a. This number a is called a quasi-coincidence value corresponding to the polynomial solutions y=y(x).The number of all solutions in (5) may be infinitely many, or finitely many, or not solvable.Since there may have many solutions corresponding to the numbera, for convenience, we use the following notations to represent different situations: (1)
QcsF, the set of all quasi-coincidence solutions satisfying (5),(2)
QcvF, the set of all quasi-coincidence values satisfying (5),(3)
QcsF(a), the set of all quasi-coincidence solutions y(x) corresponding to a fixed quasi-coincidence value a. Evidently,
(6)QcsF=⋃a∈QcvF‍QcsF(a),(7)QcsF(a)⋂‍QcsF(b)=∅
for any a≠b in QcvF. Moreover, for each a∈ℝ, the cardinal number of QcsF(a), denoted by |QcsF(a)|, satisfies the following condition:
(8)|QcsF(a)|≤degyF(x,y).In Section2, we derive some properties of quasi-coincidence solutions of F(x,y). If (5) has infinitely many quasi-coincidence solutions, the concerned properties are described in Section 3.Throughout the paper, we denote the polynomial function by(9)F(x,y)=as(x)ys+as-1(x)ys-1+⋯+a1(x)y+a0(x)=∑i=0s‍ai(x)yi.


## 2. Auxiliary Lemmas

For convenience, we explain some interesting properties of quasi-coincidence point solutions as the following lemmas. Throughout this paper, we consider (5) for polynomial function (9) and arbitrary polynomial f(x) in ℝ[x].Lemma 3.
Lety1(x)∈QcsF(a), y2(x)∈QcsF(b), a≠b in QcvF. Then,
(10)y1(x)-y2(x)=dp(x)forsomed∈ℝ,
and this p(x) is divisible f(x); that is, p(x)∣f(x).Proof.
Sincey1(x)≠y2(x) in QcsF correspond to a≠b in QcvF, respectively, thus
(11)F(x,y1(x))=af(x),F(x,y2(x))=bf(x).
Subtracting the above two equations and using binomial formula, it yields
(12)(a-b)f(x)=F(x,y1(x))-F(x,y2(x))=as(x)[y1s(x)-y2s(x)]+as-1(x)m×[y1s-1(x)-y2s-1(x)]m+⋯+a1(x)[y1(x)-y2(x)]=[y1(x)-y2(x)]m×[as(x)Gs(y1(x),y2(x))mmm+as-1(x)Gs-1(y1(x),y2(x))mmm+⋯+a1(x)]=[y1(x)-y2(x)]Q(x,y1(x),y2(x)),
where Gj(y1(x),y2(x))=y1j-1(x)+y1j-2(x)y2(x)+⋯+y2j-1(x), for j=s,s-1,…,2,1. Evidently, the factor y1(x)-y2(x) is divisible to the term (a-b)f(x).
Sincea≠b,
(13)y1(x)-y2(x)=dp(x)
for a real number d∈ℝ and some factor p(x) of f(x).In Lemma3, the difference of any two distinct quasi-coincidence solutions corresponding to distinct values is a factor of f(x). Thus, we define a class of this factor as follows.Notation. (i) Let p(x) be a factor of f(x), and we denote Φ(p(x))={αp(x):α∈ℝ}.(ii) Lety(x) be an arbitrary polynomial in ℝ[x], and we denote y(x)+Φ(p(x))={y(x)+αp(x):α∈ℝ}.It is obvious that for anyy(x), p(x)∈ℝ[x], then the cardinal number
(14)|y(x)+Φ(p(x))|=∞.For convenience, we explain the relations ofQcsF and Φ(p(x)) in the following lemma.Lemma 4.
Lety(x)∈QcsF(a) for some a∈ℝ, then
(15)QcsF=QcsF(a)m⋃(⋃p(x)∣f(x)‍{(y(x)+Φ(p(x)))∩QcsF}).Proof.
For anyy1(x)∈QcsF∖QcsF(a), then y1(x)∈Qcs(b) for some b∈QcvF. By Lemma 3, we have
(16)y1(x)-y(x)∈Φ(p(x))
for some factor p(x) of f(x). Then,
(17)y1(x)∈⋃p(x)∣f(x)‍{y(x)+Φ(p(x))}.
That is,
(18)QcsF⊆QcsF(a)⋃‍(⋃p(x)∣f(x)‍{y(x)+Φ(p(x))}).
Moreover, by (6), QcsF(a)⊆QcsF, then it follows that
(19)QcsF=QcsF(a)⋃(⋃p(x)∣f(x)‍{(y(x)+Φ(p(x)))∩QcsF}).We will use the definitions of “the pigeonhole principle;” it could concert to Grimaldi [6], and the relation can be explained as the following lemma.Lemma 5.
Suppose that the cardinal number|QcsF|=∞. For any y(x)∈QcsF, there exists a factor p(x) of f(x) such that the cardinal number
(20)|(y(x)+Φ(p(x)))∩QcsF|=∞.Proof.
Lety(x)∈QcsF, then y(x)∈QcsF(a) for some a∈ℝ. Since |QcsF|=∞ and |QcsF(a)|≤s, by Lemma 4, we obtain
(21)|⋃p(x)∣f(x)‍({y(x)+Φ(p(x))}∩QcsF)|=∞,
it yields
(22)∑p(x)∣f(x)‍|(y(x)+Φ(p(x)))∩QcsF|=∞.
Moreover, the number of all factorp(x) of f(x) is at most 2degf(x), by pigeonhole’s principle, it leads to
(23)|(y(x)+Φ(p(x)))∩QcsF|=∞
for some factor p(x) of f(x).In order to know if the intersection of two sets still has infinite solutions, we state the following result to give an explanation.Lemma 6.
Suppose that the cardinal number|QcsF|=∞, for any y1(x)≠y2(x)∈QcsF, there exist some factors p1(x) and p2(x) of f(x) such that
(24)|(y1(x)+Φ(p1(x)))∩(y2(x)+Φ(p2(x)))∩QcsF|=∞.Proof.
Lety1(x)∈QcsF and since |QcsF|=∞, by Lemma 5, there exists a factor p1(x) of f(x) such that
(25)|(y1(x)+Φ(p1(x)))∩QcsF|=∞.
Moreover, y2(x)∈QcsF, by Lemma 4,
(26)QcsF⊆QcsF(b)⋃‍(⋃p2(x)∣f(x)‍{y2(x)+Φ(p2(x))})
for some constant b∈ℝ and some factor p2(x) of f(x). Thus,
(27)({y1(x)+Φ(p1(x))}∩QcsF)m⊆QcsF⊆QcsF(b)⋃‍(⋃p2(x)∣f(x)‍{y2(x)+Φ(p2(x))}).
Since |QcsF(b)|≤s and the number of all factor to f(x) is at most 2degf(x), by pigeonhole’s principle and (25), we have
(28)|(y2(x)+Φ(p2(x)))∩QcsF|=∞,|(y1(x)+Φ(p1(x)))∩(y2(x)+Φ(p2(x)))∩QcsF|=∞
for some factor p2(x) of f(x).Up to now, we have not shown that the factorp(x) is uniquely existed. Eventually, if the number of all solutions is infinitely many, then the factor p(x) of f(x) is unique up to the choice of the solution y(x).Lemma 7.
Assume that the cardinal number|QcsF|=∞, then for any y1(x), y2(x)∈QcsF, one has
(29)y1(x)-y2(x)=λp(x)
for some constant λ∈ℝ and some factor p(x) of f(x) (this p(x) is independent to the choice of y1(x) and y2(x)).Proof.
Lety1(x)≠y2(x)∈QcsF, by Lemma 6, we have
(30)|(y1(x)+Φ(p1(x)))∩(y2(x)+Φ(p2(x)))∩QcsF|=∞
for some factors p1(x), p2(x) of f(x).
Letg1(x)≠g2(x)∈(y1(x)+Φ(p1(x)))∩(y2(x)+Φ(p2(x)))∩QcsF-{y1(x),y2(x)}, then
(31)g1(x)∈y1(x)+Φ(p1(x)),g2(x)∈y1(x)+Φ(p1(x)),g1(x)∈y2(x)+Φ(p2(x)),g2(x)∈y2(x)+Φ(p2(x)).
By Lemma 6, it yields
(32)g1(x)-y1(x)=λ1p1(x),g2(x)-y1(x)=λ2p1(x),g1(x)-y2(x)=λ3p2(x),g2(x)-y2(x)=λ4p2(x)
for some constants λ1, λ2, λ3, and λ4∈ℝ, and consequently
(33)g2(x)-g1(x)=(g2(x)-y1(x))-(g1(x)-y1(x))=(λ2-λ1)p1(x),g2(x)-g1(x)=(g2(x)-y2(x))-(g1(x)-y2(x))=(λ4-λ3)p2(x).
This implies that
(34)(λ2-λ1)p1(x)=(λ4-λ3)p2(x)
and p1(x)=p2(x). Therefore,
(35)y1(x)-y2(x)=(g1(x)-y2(x))-(g1(x)-y1(x))=(λ3-λ1)p1(x).
Consequently, the factor p(x) is uniquely existed.By the above preparations, at first we consider the polynomial functionF(x,y) with degyF=1 as the form
(36)F(x,y)=a1(x)y+a0(x).
Then, we consider the theorem of problem as
(37)F(x,y)=af(x).
In the following theorem, we integrate the above type as follows.Theorem 8.
LetF(x,y) be a polynomial function with degyF=1 as the form F(x,y)=a1(x)y+a0(x)=af(x) for some a∈ℝ (where polynomial function f(x) is given). If the cardinal number |QcvF|≥2(=degyF+1), then (i)
a1(x) is some factor of f(x),(ii)
any solution of (5) is of the form:
(38)y(x)=-a0(x)a1(x)+λp(x)
for some λ∈ℝ and some factor p(x) of f(x),(iii)
the cardinal number|QcsF|=∞.Proof.
Since|QcvF|≥2, we see that there are two distinct quasi-fixed values a, b∈QcvF corresponding to two distinct solutions y1(x), y2(x) in QcsF such that
(39)F(x,y1(x))=af(x),F(x,y2(x))=bf(x).
(i) It follows that(40)F(x,y1(x))=a1(x)y1(x)+a0(x)=af(x),(41)F(x,y2(x))=a1(x)y2(x)+a0(x)=bf(x).
By (40)-(41), we get
(42)a1(x)(y1(x)-y2(x))=(a-b)f(x).
It follows that a1(x) must be a factor of f(x) and
(43)y1(x)-y2(x)=(a-b)f(x)a1(x)∈ℝ[x].
(ii) By (40), we have
(44)a0(x)=af(x)-a1(x)y1(x)(45)=a1(x)(af(x)a1(x)-y1(x)).
Thus, (i) and (45) imply a1(x)∣a0(x), and by (44), y1(x) can be written as
(46)y1(x)=af(x)-a0(x)a1(x).
Moreover, we derive
(47)F(x,y)=a1(x)y+a0(x)=a1(x)(y-y1(x))+(a1(x)y1(x)+a0(x))by(40)=a1(x)(y-y1(x))+af(x).
For any y(x)∈QcsF, we have
(48)F(x,y(x))=a~f(x)
for some a~∈ℝ. By (47) and (48), it follows that
(49)a1(x)(y(x)-y1(x))+af(x)=a~f(x).
Hence,
(50)y(x)-y1(x)=(a~-a)f(x)a1(x).
Then,
(51)y(x)=y1(x)+(a~-a)f(x)a1(x)by(46)=af(x)-a0(x)a1(x)+(a~-a)f(x)a1(x)=-a0(x)a1(x)+a~f(x)a1(x).
Therefore,
(52)y(x)=-a0(x)a1(x)+λp(x)forsomefactorp(x)off(x).
(Note that this p(x)=f(x)/a1(x) is only dependent on the choice of F(x,y) and f(x).)
(iii) Actually in (ii), for anyλ∈ℝ, y(x)=-a0(x)/a1(x)+λf(x)/a1(x) is also a quasi-coincidence solution for F(x,y). The reason is
(53)F(x,y(x))=a1(x)y(x)+a0(x)=a1(x)(-a0(x)a1(x)+λf(x)a1(x))+a0(x)=λf(x).
This shows that (*) has infinitely many solutions (i.e., |QcsF|=∞).Remark 9.
Notice that in the case ofdegyF=1 and |QcsF|<∞, the number of all quasi-coincidence values cannot be larger than 1. Otherwise, it will contract the result of Theorem 8; the case (iii) means that “|QcvF|≥2, and then |QcsF|=∞”.


## 3. Main Theorems

In this section, we consider (5) for polynomial function F(x,y) in (9); that is,
(54)F(x,y)=∑i=0s‍ai(x)yiwiths≥2.
A given polynomial function f(x) in ℝ[x] and F(x,y) has at least s+1 distinct quasi-coincidence solutions satisfying some conditions, that is, y1(x), y2(x), y3(x),…, ys+1(x),…. According to the above assumptions, we could derive the following theorem.Theorem 10.
Suppose that the cardinal number|QcsF|=∞ and for each y(x)∈QcsF can be represented as the form
(55)y(x)=y1(x)+λp(x),λ∈ℝ
for some y1(x), p(x)∈ℝ[x]. Then, ps(x)∣f(x), and so the polynomial F(x,y) can be represented as
(56)F(x,y)=∑i=0s‍cif(x)pi(x)(y-y1(x))i
for constants ci∈ℝ.Proof.
Letyi(x) be distinct quasi-coincidence solutions of F(x,y) corresponding to quasi-coincidence values ai, 1≤i≤s+1 such that
(57)F(x,yi(x))=aif(x).
Choose i=1, F(x,y1(x))=a1f(x). When y-y1(x) divides the function F(x,y), we get
(58)F(x,y)=(y-y1(x))F1(x,y)+a1f(x),
where F1(x,y) is the quotient and a1f(x) is the remainder. From the above identity, taking y=y2(x), it becomes
(59)F(x,y2(x))=(y2(x)-y1(x))F1(x,y2(x))+a1f(x)=a2f(x).
Then,
(60)(y2(x)-y1(x))F1(x,y2(x))=(a2-a1)f(x).
By (55), y2(x)-y1(x)=λ2p(x), it yields
(61)F1(x,y2(x))=((a2-a1)λ2)f(x)p(x)=d2f(x)p(x)∈ℝ[x]ford2=(a2-a1)λ2.
Hence,
(62)F1(x,y)=(y-y2(x))F2(x,y)+d2f(x)p(x).
Continuing this process from i=2 to s-1, we obtain
(63)Fi(x,y)=(y-yi+1(x))Fi+1(x,y)+di+1f(x)pi(x)
for some di+1∈ℝ, i=1,2,…,s-1. Finally, we could get
(64)Fs-1(x,y)=(y-ys(x))Fs(x)+dsf(x)ps-1(x).Fs(x) does not contain the variable y since degyF=s. By the assumption (57), F(x,ys+1(x))=as+1f(x). It follows that
(65)Fs(x)=λf(x)ps(x)∈ℝ[x]forsomeconstantλ∈ℝ.
Consequently,
(66)F(x,y)=(y-y1(x))F1(x,y)+a1f(x)=(y-y1(x))×((y-y2(x))F2(x,y)+d2f(x)p(x))+a1f(x)=…=(y-y1(x))(+d2f(x)p(x)(y-y2(x))mmmmmmmm×(+dsf(x)ps-1(x))⋯(dsf(x)ps-1(x)(y-ys(x))Fs(x)mmmmmmmmmmmm+dsf(x)ps-1(x))⋯)mmmmmmmm+d2f(x)p(x))+a1f(x)=(y-y1(x))(+d2f(x)p(x)(y-y2(x))mmmmmmmm×(⋯((y-ys(x))λf(x)ps(x)mmmmmmmmmmmm+dsf(x)ps-1(x))⋯)mmmmmmmm+d2f(x)p(x))+a1f(x).
By (55), we have yi(x)=y1(x)+λip(x),  i=2,3,…,s+1. Then, F(x,y) can be expanded to a power series in the expression:
(67)F(x,y)=(y-y1(x))×(+d2f(x)p(x)(y-y1(x)-λ2p(x))m×(⋯((y-y1(x)-λsp(x))λf(x)ps(x)mmmmm+dsf(x)ps-1(x))⋯)m+d2f(x)p(x))+a1f(x)=∑i=0s‍cif(x)pi(x)(y-y1(x))i
for some real numbers cj, j=0,1,…,s. Moreover, the leading coefficient of F(x,y), cs(f(x)/ps(x)) is contained to ℝ[x], and it follows ps(x)∣f(x).Conversely, ifF(x,y) is expressed as in Theorem 10, then the cardinal number |QcsF|=∞, this is the same as the sufficient conditions.Theorem 11.
The following two conditions are equivalent:(i)
F(x,y)=∑i=0sci(f(x)/pi(x))(y-y1(x))i for some y1(x)∈ℝ[x], p(x) is a factor of f(x), and ci∈ℝ for i=0,1,…,s,(ii)
|QcsF|=∞.(In fact, if|QcsF|=∞, then |QcsF|= the cardinal number of ℝ.)Proof.
(i)⇒(ii) Suppose that (i) holds. Then,
(68)F(x,y1(x)+λp(x))=∑i=0s‍cif(x)pi(x)(λp(x))i=(∑i=0s‍ciλi)f(x)=apm(x)fora=∑i=0s‍ciλi∈ℝ.
This means that y1(x)+λp(x)∈QcsF for each λ∈ℝ. It follows that the cardinal |QcsF|=∞.
(ii)⇒(i) For any y(x),y1(x)∈QcsF, since |QcsF|=∞ and by Lemma 7, we obtain
(69)y(x)-y1(x)=dp(x)
for some factor p(x) of f(x). By Theorem 10, we have
(70)F(x,y)=∑i=0s‍cif(x)pi(x)(y-y1(x))i
for some y1(x), p(x)∈ℝ[x], and ci∈ℝ for i=0,1,…,s.If theF(x,y) can be represented as the form of (71) in the following lemma, then any quasi-coincidence solution can be determined.Lemma 12.
Suppose that(71)F(x,y)=∑i=0s‍cif(x)pi(x)(y-y(x))i,
where y(x)∈ℝ[x], ci∈ℝ, i=0,1,…,s and p(x) is a factor of f(x). Then, h(x)∈ℝ[x] is a quasi-coincidence solution of F(x,y), if and only if
(72)h(x)=y(x)+dp(x)forsomed∈ℝ.Proof.
At first, we assume thath(x)∈ℝ[x] is a quasi-coincidence solution of F(x,y), and we consider
(73)F(x,y(x))=∑i=0s‍cif(x)pi(x)(y(x)-y(x))i=c0f(x).
This means y(x)∈QcsF.
By Theorem11,
(74)F(x,y)=∑i=0s‍cif(x)pi(x)(y-y(x))i,then|QcsF|=∞.
It follows from Lemma 7 that for any quasi-coincidence solution h(x), we obtain
(75)h(x)=y(x)+dp(x)forsomed∈ℝ.
Conversely, suppose h(x)=y(x)+dp(x) for some factor p(x) of f(x) and some constant d∈ℝ. Substituting this h(x) as y in (71), we have
(76)F(x,h(x))=F(x,y(x)+dp(x))=∑i=0s‍cif(x)pi(x)(dp(x))i=(∑i=0s‍cidi)f(x).
Therefore, h(x)∈QcsF.
Note that not all polynomial functionsF(x,y) can be written as (71). Actually, almost all F(x,y) are expressed as the form of the next theorem. In that situation, any solution can be written as the next form (⋆) if the cardinal number |QcsF| is in_nitely many in this theorem.Theorem 13.
LetF(x,y) be a polynomial function with
(77)F(x,y)=as(x)ys+as-1(x)ys-1+⋯+a0(x),
and f(x) a polynomial. If the cardinal number |QcsF| is infinitely many, then for each quasi-coincidence point solution of (5) must be of the form
(⋆)-as-1(x)sas(x)+λp(x)
for arbitrary λ∈ℝ, where p(x)=c(f(x)/as(x))1/s is a factor of f(x) and c is a constant.Proof.
Assume|QcsF|=∞. By Theorem 11, we have
(78)F(x,y)=as(x)ys+as-1(x)ys-1+⋯+a0(x)=∑i=0s‍cif(x)pi(x)(y-y(x))i
for some ci∈ℝ, and y(x)∈QcsF. Comparing the coefficients of ys and ys-1 in both sides, we get
(79)as(x)=csf(x)ps(x),(80)as-1(x)=-sas(x)y(x)+cs-1f(x)ps-1(x).
Consequently, by (79) and (80), we get
(81)ps(x)=csf(x)as(x),y(x)=cs-1scsp(x)-as-1(x)sas(x)∈ℝ[x].
By Lemma 12 and (81), for any d∈ℝ, we have that any quasi-coincidence solution is represented by
(82)y(x)+dp(x)=cs-1scsp(x)-as-1(x)sas(x)+dp(x)=-as-1(x)sas(x)+(d-cs-1scs)p(x)=-as-1(x)sas(x)+λp(x),
where p(x)=(cs)1/s(f(x)/as(x))1/s (note that since d is arbitrary, then λ is arbitrary).
This completes the proof.Finally, we provide two examples. Example1 explains the case of all cardinal number |QcsF|=4.Example 1.
Let(83)F(x,y)=(x2+x+1)y2-x6-3x5-6x4-7x3-10x2-7x-5,f(x)=(x2+x+1)2.
Then,
(84)F(x,y)=(x2+x+1)×[y2-(x2+x+1)2-4],degyF=s=2.
This polynomial The polynomial equation F(x,y)=a(x2+x+1)2 for some a∈ℝ has exactly 4=(s+2) quasi-coincidence solutions as follows:
(85)F(x,x2+x+3)=4(x2+x+1)2,F(x,-x2-x-3)=4(x2+x+1)2,F(x,x2+x-1)=-4(x2+x+1)2,F(x,-x2-x+1)=-4(x2+x+1)2.The next example explains that the number of all quasi-coincidence solutions of (5) is infinitely many.Example 2.
Letx∈ℝ, f(x)=x4(x-1)4, and
(86)F(x,y)=a3(x)y3+a2(x)y2+a1(x)y+a0(x)=x(x-1)y3+0y2+x3(x-1)3y+0.
We will solve all quasi-coincidence solutions of F(x,y)=ax4(x-1)4 for some a∈ℝ. This polynomial function has at least 6(≥s+3, since s=3) quasi-coincidence solutions as follows:
(87)F(x1,x2,x2-x)=2x4(x-1)4,F(x1,x2,2x2-2x)=10x4(x-1)4,F(x1,x2,-x2+x)=-2x4(x-1)4,F(x1,x2,-2x2+2x)=-10x4(x-1)4,F(x1,x2,x22-x2)=58x4(x-1)4,F(x1,x2,-x22+x2)=-58x4(x-1)4.
In fact, we have |QcsF|=∞, and by (79), we obtain
(88)p(x)=c(f(x)as(x))1/s=c(x4(x-1)4x(x-1))1/3=cx(x-1),
for some real number c.
By Theorem13, any quasi-coincidence solution is written as
(89)-a2(x)sa3(x)+λp(x)=03x(x-1)+λcx(x-1)=μx(x-1),
where μ=λc∈ℝ is arbitrary. This shows that the quasi-coincidence (point) solutions have cardinal |QcsF|=∞.We would like to provide one open problem as follows.Further Development. For a real-valued polynomial function F:ℝ×ℝ→ℝ. Can we find all rational quasi-coincidence solutions y=b(x)/a(x) with coprime polynomials a(x), b(x)∈ℝ[x] to satisfy
(90)F(x,y)=af(x)
for some polynomials f(x)∈ℝ[x]?



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*Source: 959464-2013-09-09.xml*