Infinitely Many Quasi-Coincidence Point Solutions of Multivariate Polynomial Problems

Abstract

Let be a real-valued polynomial function of the form where the degree of in is greater than . 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 of (). In this paper, we prove that the number of all solutions in () does not exceed provided those solutions are of finitely many exist, if all solutions are of infinitely many exist, then any solution is represented as the form where is arbitrary and is also a factor of , provided the equation () has infinitely many quasi-coincidence (point) solutions.

1. Introduction

In 1987, Lenstra [1] researched a polynomial function     ( is an algebraic number) and attempted to search the factorization of . Continuing his job, many scientists tried to find the roots of the polynomial equations (cf. [2–6]). Later, many authors also studied fixed point theory and fixed coincidence theory (cf. [7–11]). Recently, Lai and Chen ([12–15]) research the quasi-fixed (point) polynomial problem; they assumed a polynomial function and solved 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 (Lai and Chen [12]). A polynomial function satisfying (1) is called a quasi-fixed solution corresponding to some real number . This number is called a quasi-fixed value corresponding to the polynomial solutions .

Moreover, Chen and Lai [16] extended (1) to a more general coincidence (point) problem in which the is replaced by the irreducible polynomial power , where is an arbitrary polynomial. Then we restate (1) as the following equation: It is a new development coincidence point-like problem. We call the polynomial solution for (3) a quasi-coincidence (point) solution. Precisely, we give the following definition like Definition 1.

Definition 2 (Chen and Lai [16]). A polynomial function satisfying (3) 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 .

Furthermore, we consider a multivariate polynomial function and extend (3) as a more general coincidence (point) problem in which the is replaced by throughout this paper, where is a nonzero arbitrary polynomial in . Then we restate (3) as the following equation: Thus, we can give some definitions like Definition 2 as follows.

Definition 3. A polynomial function satisfying (4) 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 (4) may be infinitely many, finitely many, or not solvable. In this paper, we solve all solutions of (4) if the number is infinitely many. Moreover, we provide an upper bound for the number of all solutions if the number is finitely many.

In Section 2, we derive some properties of quasi-coincidence solutions. If (4) has infinitely many quasi-coincidence solutions, the form of will be described in Section 3. In the last section, we solve all solutions if (4) has infinitely many solutions.

2. Preliminaries

For convenience, we denote the polynomial function by throughout this paper and since there may exist many solutions corresponding to the same number , we use the similar notations like (Definition  2, [11]) to represent them.

Notation 1. (1) , the set of all solutions satisfying equation (4), the solution in is also called a quasi-coincidence solution in (4) (like Definition 2).
(2) , the set of all solutions satisfying equation (4), the solution in is also called a quasi-coincidence value in (4) (like Definition 2).
(3) , the set of all quasi-coincidence solutions corresponding to a quasi-coincidence value .
(4) For each , we denote as the cardinal number of .

Evidently, by Notation 1, we have the following lemma.

Lemma 4. (i) ;
(ii) for any in ;
(iii) for any ;
(iv) for any .

Proof. (i) is obvious.
Conversely, for any , by Notation 1(1), we have for some . This means and we obtain (ii) Let in ; if there exists such that by Notation 1(3), we have This leads a contradiction to and we have .
(iii) For each , the number of all solutions to the polynomial equation is at most ; then the result is obtained.
(iv) By (i), we have It follows that

In the following lemma, we explain some interesting properties of the relations of quasi-coincidence point solutions. Throughout this paper, we consider (4) for polynomial function (5) and nonzero arbitrary polynomial in .

Lemma 5. Let the cardinal number and in . Then for any and , one has and this is a factor of , that is, .

Proof. Since , correspond to , , respectively, we have Subtracting (14) from (13) and using binomial formula, it yields that where for , . Evidently, the factor is divisible to the term and since , we obtain for some real number and factor of .

In Lemma 5, the difference of any two distinct quasi-coincidence solutions corresponding to distinct values is a factor of . Thus we may define a class of those factors in the following.

Notation 2. (i) Denote .
(ii) Let be an arbitrary polynomial in , and we denote .
(iii) .

If , correspond to distinct quasi-coincidence values, by Lemma 5 and Notation 2, we have

Since the number of all factors to is at most , by the definitions of “the pigeonhole principle” in [17], we have the following results.

Lemma 6. Suppose that Then there exists and is a factor of such that

Proof. Since , by (18), there exists , such that for some factor of , for . Moreover, we have that the number of all factors to is at most . By “the pigeonhole principle,” there exists a subset such that for and some factor of (this is independent of the choice of ). Thus

For convenience, we explain the relations of and in the following lemma.

Lemma 7. Let for some . Then for some factor of .

Proof. For any , then for some . By Lemma 5, we have for some factor of . Then and it follows that Moreover, by Lemma 4(i), ; then we obtain

In order to let the number of all elements in the intersection of sets and be large enough, we find a lower bound for in the following theorem.

Theorem 8. Suppose that the cardinal number Then for any , there exist two factors and of such that

Proof. Let and by assumption and by Lemma 6, there exists a factor of such that This implies that Moreover, for any , we have for some constant and it follows that Canceling both sides of the above inequality by “”, it follows that this implies that By the pigeonhole’ principle and since the number of all factors to is at most , we have for some factor of . Thus we obtain

Up to now, we have not shown that the factor uniquely existed eventually. In the following theorem, we would show the uniqueness property for the factor of if the number of all quasi-coincidence values is large enough.

Theorem 9. Assume that the cardinal number Then for any , one has where and is a factor of (this is independent of the choice of and ).

Proof. Let ; by Theorem 8, we have for some factors and of . There exists , such that This implies that By Notation 2(ii), it yields that for some constants , , , and and consequently This implies that and . Therefore, and this means that the factor of is uniquely determined independent of the choice of and .

Corollary 10. Assume that the cardinal number Then for any , there exists such that for some   ( are independent of the choice of ).

Proof. By Lemma 4(iv), we have By assumption, it follows that Dividing both sides of the above equation by , we get If , for any , by Theorem 9, we have for some factor of .

3. The Type of If the Number of All Quasi-Coincidence Solutions Is Infinitely Many

In this section, we consider (4) for polynomial function in (5); that is, let and we assume that has at least distinct quasi-coincidence solutions satisfying some conditions, that is, in the following theorem. According to the above assumptions, we could derive the following result.

Theorem 11. Suppose that the cardinal number and for each can be represented as the form for some and , for . Then and 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, take , and it becomes Then By (56), , it yields that Hence Continuing this process from to , we obtain for some , . Finally, we could get does not contain the variable since . By the assumption (58), . It follows that Consequently, By (56), 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 .

In the above theorem, if there exist at least quasi-coincidence solutions with some relations, then has a fixed type. In the following theorem, if has a fixed type expressed as in Theorem 11, then the cardinal number .

Theorem 12. The following three conditions are equivalent:(i) for some , some factor of and , ;(ii);(iii). (In fact, if , then the cardinal number of ).

Proof. Suppose that (i) holds. Let for any constant ; we have This means that   for all and we obtain It follows that the cardinal number .
can be obtained obvious from Lemma 4(iv).
For any , by Theorem 9, we have for some fixed factor of and by Theorem 11, we obtain for some and , .

Corollary 13. If the number of all quasi-fixed solutions is finitely many, the number of all quasi-fixed values does not exceed an integer . Actually,

Proof. By the contrapositive of Theorem 12, we have “if , then .” Hence the number of all quasi-fixed values is at most ; that is, .

Corollary 14. If the number of all quasi-fixed solutions is finitely many, the number of all quasi-fixed solutions does not exceed

Proof. By Lemma 4(iv), we have for any

4. Main Theorems and Some Corollaries

If the can be represented as the form (57), then any quasi-coincidence solution can be formed in this section.

Lemma 15. Let be represented as in (57). Then is a quasi-coincidence solution of if and only if for some and some factor of .

Proof. Since we let , and then this means .
By Theorem 12, Assume that is a quasi-coincidence solution of and by Corollary 10, we obtain that for any quasi-coincidence solution , we have Conversely, suppose , for some factor of and . Substituting this as in (57), we have Therefore .

Note that not any polynomial function can be written as (57). Actually, almost all are expressed as the form of the next theorem. In this situation, any solution can be written as the next form in this theorem under some condition.

Theorem 16. Let be a polynomial function with and a polynomial. If the cardinal number is infinitely many, then each quasi-coincidence solution of (4) must be of the form for arbitrary , where is a factor of .

Proof. Assume . By Theorem 12, we have for some , , and . Comparing the coefficients of and in both sides of the above equation, we get Consequently, by (84), we get By Lemma 15, 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.

Corollary 17. Let be a polynomial function with If the cardinal number of equation is infinitely many, then the leading coefficient must be a real number and each quasi-coincidence point solution must be of the form for arbitrary .

Corollary 18. Let be a polynomial function with If the cardinal number of equation is infinitely many, then the leading coefficient must be a real number and each quasi-coincidence point solution must be of the form for arbitrary and some .

Proof. Assume that there exist infinitely many solutions; by Theorem 16, any solution of (4) has the form for arbitrary and , and then is a factor of . This means that or for some constant , if ; this implies , and this leads a contradiction. So we have , for some ; then and any solution (91) is represented as for arbitrary and some .

Finally, we provide one example to explain Theorem 16.

Example 19. Let , , and Can we solve all quasi-fixed solutions of ? This polynomial function has exactly 5(≥, since quasi-fixed solutions as follows: In fact, by Theorem 16, we can find any quasi-coincidence solution written as where is arbitrary and . This shows the quasi-coincidence (point) solutions have cardinal .
In practice, we have no idea to check the number of this equation is infinitely many or finitely many. But we provide an easy method to solve all solutions if the number of all solutions is infinitely many in this paper. Thus, we can solve those solutions directly and check whether those solutions are the solutions of and give an example in the following.

Example 20. Let , , and We will solve all quasi-fixed solutions of if the number of all solutions is infinitely many.
By Theorem 16, we can find that any quasi-coincidence solution can be written as where is arbitrary and . We let and calculate and obtain This means that the quasi-coincidence (point) solutions have cardinal .

We would like to provide one open problem as follows.

Further Development. Let be a quotient field. Consider a quotient-valued polynomial function Can we find all quasi-coincidence solutions to satisfy for some polynomials by a co-NP hardness algorithm?