The Number of Solutions of Certain Equations over Finite Fields

: Let s be a positive integer, p be an odd prime, s q p  , and let q F be a finite field of q elements. Let q N be the number of solutions of the following equations:

and let q F be a finite field of q elements.Let q N be the number of solutions of the following equations: In this paper, we find formulas for q N when there is a positive integer l such that dD|( / ), lcm And we determine q 0 Introduction Let p be an odd prime.Let q F be a finite field of q elements with , s p q  1 s ≥ and * q F denote the set of all the nonzero elements of q F .Let N(f = b) denote the number of solutions of the equation 1 2 ( , , , )  where 1 2 ( , , , ) 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.Markoff-Hurwitz-type equations belong to the following type of the Diophantine equations where n, a are positive integers and 3 n ≥ .This type of equations was first studied by Markoff [1] for the case 3, 3 n a   .More generally, these equations were studied by Hurwitz [2] .Recently, Baoulina [3][4][5] studied the generalized Markoff-Hurwitz-type equations where 2, , , [8] .Song and Chen [9] presented the formulas for the number of solutions of the following equations q a F  Hu and Li [10] consider the rational points of the further generalized Markoff-Hurwitz-type equations of the form over the finite field q F under some certain cases, where In this paper, we consider the number of solutions of the following equations over the finite field q F under some other restrictions, .
integer.Denote by q N the number of solutions of (2) in n q F .Our main result is the following theorem.

Theorem 1
Suppose that 1 gcd( , , ) , and there is a positive integer l such that | ( 1) l dD p  , with l as chosen minimal.Then 2 | l s 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.

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 -th Lemma 2 [10] Let 1 2 , , , r t t t  be positive integers and 1 2 gcd( , , , , 1) . Then for any elements * , , ( 1) , if is a -th power in 0, otherwise The following two lemmas are the main results in Ref. [6] and fundamental for our results.
Lemma 3 [6] Let 2. n ∨ Suppose that there is a positive integer l such that 2 | l s and | ( 1) and there is a positive integer l such that | ( 1) ( 1) ( , , ) ( 1) , if is a -th power but not a -th power in 0, if is not a -th power in 2 Proof of Theorem 1 In this section, we give the proof of Theorem 1. Proof of Theorem 1 Let q N (resp.q N  ) denote the number of the solutions of the equations ).Clearly, one has Then we can solve the problem in two cases.One is and the other one is ( 1) Using the assumption there is a positive integer l such that 2 | l s and | ( 1) l dD p  .Thus, by (4) and Lemma 3 , , , 1) .