Issue 
Wuhan Univ. J. Nat. Sci.
Volume 27, Number 1, March 2022



Page(s)  49  52  
DOI  https://doi.org/10.1051/wujns/2022271049  
Published online  16 March 2022 
Mathematics
CLC number: O156
The Number of Solutions of Certain Equations over Finite Fields
^{1} School of Mathematics and Statistics, Nanyang Institute of Technology, Nanyang 473004, Henan, China
^{2} College of Computer and Information Science, Southwest University, Chongqing 400715, China
^{3} School of Mathmatics and Computer, Panzhihua University, Panzhihua 617000, Sichuan, China
Received: 2 November 2021
Let s be a positive integer, p be an odd prime, , and let be a finite field of q elements. Let be the number of solutions of the following equations: over the finite field , with and are positive integers. In this paper, we find formulas for when there is a positive integer l such that dD() , where And we determine explicitly under certain cases. This extends MarkoffHurwitztype equations over finite field.
Key words: finite field / rational point / diagonal equation / MarkoffHurwitztype equations
Biography: HU Shuangnian, male, Ph. D., Associate professor, research direction: number theory . Email: hushuagnian@163.com
Foundation item: Supported by the National Natural Science Foundation of China (12026224)
© Wuhan University 2022
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
0 Introduction
Let p be an odd prime. Let be a finite field of q elements with and denote the set of all the nonzero elements of . Let N(f = b) denote the number of solutions of the equation in where is a polynomial in and . That is,Studying the value of N(f = b) is one of the main topics in finite fields. Generally speaking, it is nontrivial to give the formula for N(f = b). Finding the explicit formula for N(f = b) under certain condition has attracted lots of authors for many years.
MarkoffHurwitztype equations belong to the following type of the Diophantine equationswhere n, a are positive integers and . This type of equations was first studied by Markoff^{[1]} for the case . More generally, these equations were studied by Hurwitz^{[2]}.
Recently, Baoulina^{[35]} studied the generalized MarkoffHurwitztype equationswhere and are positive integers satisfying for and . Baoulina^{[46]} and Pan et al^{[7]} considered the further generalized Markoff Hurwitztype equations of the form:(1)where are positive integers, , for . The special case (1) of is investigated by Cao^{[8]}. Song and Chen^{[9]} presented the formulas for the number of solutions of the following equationsover the finite field under some certain restrictions, where for and
Hu and Li^{[10]} consider the rational points of the further generalized MarkoffHurwitztype equations of the formover the finite field under some certain cases, where , and are positive integers, , for
In this paper, we consider the number of solutions of the following equations(2)over the finite field under some other restrictions, where , , are positive integers. In what follows, we always let
For any positive integers we let denote the number of of integers with such that is an integer. Denote by the number of solutions of (2) in . Our main result is the following theorem.
Theorem 1 Suppose that and there is a positive integer l such that , with l as chosen minimal. Then and
This paper is organized as follows. In Section 1, we review some useful known lemmas which will be needed later. Subsequently, in Section 2, we prove Theorem 1. Some interesting applications of Theorem 1 will be provided as corollaries at the end of this paper.
1 Preliminary Lemmas
In this section, we present some useful lemmas that are needed in the proof of Theorem 1 as follows.
Lemma 1 ^{[9,11]} For any positive integer m, the number of elements of mth power in is .
Lemma 2 ^{[10]} Let be positive integers and . Then for any elements we have
The following two lemmas are the main results in Ref.[6] and fundamental for our results.
Lemma 3 ^{[6]} Let Suppose that there is a positive integer l such that and . Then
Lemma 4 ^{[6]} Suppose that and there is a positive integer l such that , with l chosen minimal. Then andwhereand
2 Proof of Theorem 1
In this section, we give the proof of Theorem 1.
Proof of Theorem 1 Let (resp. ) denote the number of the solutions of the equations with (resp. ). Clearly, one has(3)
Then we can solve the problem in two cases. One is and the other one is .
Using the assumption there is a positive integer l such that and . Thus, by (4) and Lemma 3 ,(5)
Case (ii) If , we let . Define
Note that Then from Lemma 2, we can deduce that(6)
Noting that integer l such that , with l chosen minimal. Thus for any given from Lemma 1 and Lemma 3, one has(7)
Then by (6) together with (7), we have(8)
The desired result can follow immediately from (3), (5) and (8). This ends the proof of Theorem 1.
To conclude this section, we present some corollaries. It is clear that plays a central role in Theorem 1. For any positive integers Sun and Wan^{[12]} showed that if and only if either for some j or t is odd, are pairwise coprime, and each is coprime with any odd number in , where is the set of even integers among . In Ref.[13], they also showed that if and only if are pairwise coprime, and at least (r1) of the are odd.
Therefore we can easily deduce the following corollary.
Corollary 1 Suppose that are odd, are even, are pairwise coprime, . Under the conditions of Theorem 1, we havewhereand
Let v be a positive integer. It is also known (Ref. [14], Proposition 6.17) that
Then we have the second corollary.
Corollary 2 Suppose that Under the conditions of Theorem 1, we have
Clearly, Corollaries 12 are some special cases of Theorem 1. For example, consider the further generalized MarkoffHurwitztype equation over (9)
Clearly Then we get (5, 2, 6)=30, and One can immediately conclude that (9) has 6 817 solutions in by Corollary 1.
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