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Yi-Chou Chen, Hang-Chin Lai, "New Quasi-Coincidence Point Polynomial Problems", Journal of Applied Mathematics, vol. 2013, Article ID 959464, 8 pages, 2013. https://doi.org/10.1155/2013/959464
New Quasi-Coincidence Point Polynomial Problems
Let be a real-valued polynomial function of the form , where the degree of in is greater than or equal to . For arbitrary polynomial function , , we will find a polynomial solution to satisfy the following equation: (): , where is a constant depending on the solution , namely, a quasi-coincidence (point) solution of (), and is called a quasi-coincidence value. In this paper, we prove that (i) the leading coefficient must be a factor of , and (ii) each solution of () is of the form , where is arbitrary and is also a factor of , for some constant , provided the equation has infinitely many quasi-coincidence (point) solutions.
1. Introduction and Preliminaries
Let (where is an algebraic number) be a polynomial function. Lenstra  investigated that . He solved a polynomial function and derived to find a polynomial satisfying an as a fixed point of the polynomial equation. That is, has a polynomial solution .
Recently, Lai and Chen [4, 5] extended (2) to solve to satisfy the polynomial equation as the form: where , is an irreducible polynomial in and the polynomial function is written by where denotes the degree of in .
Definition 1 (see ). A polynomial function satisfying (3) is called a quasi-fixed solution corresponding to some real number . This number is called a quasi-fixed value corresponding to the polynomial solutions .
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 the is replaced by the irreducible polynomial power throughout this paper, where is an arbitrary polynomial. Then, we restate (3) as the following equation: It is a new development coincidence point-like problem. We call the polynomial solution for (5) as a quasi-coincidence (point) solution. Precisely, we give the following definition like Definition 1.
Definition 2. A polynomial function satisfying (5) is called a quasi-coincidence (point) solution corresponding to some real number . This number is called a quasi-coincidence value corresponding to the polynomial solutions .
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 number , for convenience, we use the following notations to represent different situations: (1), the set of all quasi-coincidence solutions satisfying (5), (2), the set of all quasi-coincidence values satisfying (5), (3), the set of all quasi-coincidence solutions corresponding to a fixed quasi-coincidence value . Evidently, for any in . Moreover, for each , the cardinal number of , denoted by , satisfies the following condition:
Throughout the paper, we denote the polynomial function by
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 in .
Lemma 3. Let , , in . Then, and this is divisible ; that is, .
Proof. Since in correspond to in , respectively, thus
Subtracting the above two equations and using binomial formula, it yields
where , for . Evidently, the factor is divisible to the term .
Since , for a real number and some factor of .
In Lemma 3, the difference of any two distinct quasi-coincidence solutions corresponding to distinct values is a factor of . Thus, we define a class of this factor as follows.
Notation. (i) Let be a factor of , and we denote .
(ii) Let be an arbitrary polynomial in , and we denote .
It is obvious that for any , , then the cardinal number
For convenience, we explain the relations of and in the following lemma.
Lemma 4. Let for some , then
We will use the definitions of “the pigeonhole principle;” it could concert to Grimaldi , and the relation can be explained as the following lemma.
Lemma 5. Suppose that the cardinal number . For any , there exists a factor of such that the cardinal number
Proof. Let , then for some . Since and , by Lemma 4, we obtain
Moreover, the number of all factor of is at most , by pigeonhole’s principle, it leads to for some factor of .
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 , for any , there exist some factors and of such that
Proof. Let and since , by Lemma 5, there exists a factor of such that Moreover, , by Lemma 4, for some constant and some factor of . Thus, Since and the number of all factor to is at most , by pigeonhole’s principle and (25), we have for some factor of .
Up to now, we have not shown that the factor is uniquely existed. Eventually, if the number of all solutions is infinitely many, then the factor of is unique up to the choice of the solution .
Lemma 7. Assume that the cardinal number , then for any , , one has for some constant and some factor of (this is independent to the choice of and ).
Proof. Let , by Lemma 6, we have
for some factors , of .
Let , then By Lemma 6, it yields for some constants , , , and , and consequently This implies that and . Therefore, Consequently, the factor is uniquely existed.
By the above preparations, at first we consider the polynomial function with as the form Then, we consider the theorem of problem as In the following theorem, we integrate the above type as follows.
Theorem 8. Let be a polynomial function with as the form for some (where polynomial function is given). If the cardinal number , then (i) is some factor of , (ii)any solution of (5) is of the form: for some and some factor of ,(iii)the cardinal number .
Proof. Since , we see that there are two distinct quasi-fixed values , corresponding to two distinct solutions , in such that
(i) It follows that By (40)-(41), we get It follows that must be a factor of and
(ii) By (40), we have Thus, (i) and (45) imply , and by (44), can be written as Moreover, we derive For any , we have for some . By (47) and (48), it follows that Hence, Then, Therefore, (Note that this is only dependent on the choice of and .)
(iii) Actually in (ii), for any , is also a quasi-coincidence solution for . The reason is This shows that () has infinitely many solutions (i.e., ).
Remark 9. Notice that in the case of and , the number of all quasi-coincidence values cannot be larger than . Otherwise, it will contract the result of Theorem 8; the case (iii) means that “, and then ”.
3. Main Theorems
In this section, we consider (5) for polynomial function in (9); that is, A given polynomial function in and has at least distinct quasi-coincidence solutions satisfying some conditions, that is, , , , . According to the above assumptions, we could derive the following theorem.
Theorem 10. Suppose that the cardinal number and for each can be represented as the form for some , . Then, , and so the polynomial can be represented as for constants .
Proof. Let be distinct quasi-coincidence solutions of corresponding to quasi-coincidence values , such that Choose , . When divides the function , we get where is the quotient and is the remainder. From the above identity, taking , it becomes Then, By (55), , it yields Hence, Continuing this process from to , we obtain for some , . Finally, we could get does not contain the variable since . By the assumption (57), . It follows that Consequently, By (55), we have , . Then, can be expanded to a power series in the expression: for some real numbers , . Moreover, the leading coefficient of , is contained to , and it follows .
Conversely, if is expressed as in Theorem 10, then the cardinal number , this is the same as the sufficient conditions.
Theorem 11. The following two conditions are equivalent: (i) for some , is a factor of , and for ,(ii).
(In fact, if , then the cardinal number of .)
Proof. (i)(ii) Suppose that (i) holds. Then,
This means that for each . It follows that the cardinal .
(ii)(i) For any , since and by Lemma 7, we obtain for some factor of . By Theorem 10, we have for some , , and for .
If the can be represented as the form of (71) in the following lemma, then any quasi-coincidence solution can be determined.
Lemma 12. Suppose that where , , and is a factor of . Then, is a quasi-coincidence solution of , if and only if
Proof. At first, we assume that is a quasi-coincidence solution of , and we consider
This means .
By Theorem 11, It follows from Lemma 7 that for any quasi-coincidence solution , we obtain Conversely, suppose for some factor of and some constant . Substituting this as in (71), we have Therefore, .
Note that not all polynomial functions can be written as (71). Actually, almost all 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 is in_nitely many in this theorem.
Theorem 13. Let be a polynomial function with and a polynomial. If the cardinal number is infinitely many, then for each quasi-coincidence point solution of (5) must be of the form for arbitrary , where is a factor of and is a constant.
Proof. Assume . By Theorem 11, we have
for some , and . Comparing the coefficients of and in both sides, we get
Consequently, by (79) and (80), we get
By Lemma 12 and (81), for any , we have that any quasi-coincidence solution is represented by
where (note that since is arbitrary, then is arbitrary).
This completes the proof.
Finally, we provide two examples. Example 1 explains the case of all cardinal number .
Example 1. Let Then, This polynomial The polynomial equation for some has exactly quasi-coincidence solutions as follows:
The next example explains that the number of all quasi-coincidence solutions of (5) is infinitely many.
Example 2. Let , , and
We will solve all quasi-coincidence solutions of for some . This polynomial function has at least , since quasi-coincidence solutions as follows:
In fact, we have , and by (79), we obtain
for some real number .
By Theorem 13, any quasi-coincidence solution is written as where is arbitrary. This shows that the quasi-coincidence (point) solutions have cardinal .
We would like to provide one open problem as follows.
Further Development. For a real-valued polynomial function . Can we find all rational quasi-coincidence solutions with coprime polynomials , to satisfy for some polynomials ?
The authors wish to express their deep gratitude to Professor Tamaki Tanaka for his valuable comments on this paper and thank the referees for the very useful suggestions and remarks that contributed to the improvement of the paper.
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Copyright © 2013 Yi-Chou Chen and Hang-Chin Lai. 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.