1 Introduction
Let be a prime power and denote the finite field with elements. A polynomial over is called a permutation polynomial if the induced mapping from to itself is a bijection [15]. Permutation polynomials have been studied for several decades and have important applications in a wide range of areas such as coding theory [2, 9], combinatorial designs [4] and cryptography [18, 19].
The construction of permutation polynomials with a simple algebraic form is an interesting research problem and it has already attracted researchers’ much attention in recent years. By using certain techniques in dealing with equations or polynomials over finite fields, a number of permutation polynomials with a simple form have been obtained, the reader is referred to [3, 5, 6, 7, 8, 10, 12, 16, 23, 24, 26, 27] and the references therein. Motivated by the observation that more than half of the known permutation binomials and trinomials were constructed from Niho exponents, Li and Helleseth [12] aimed to investigate permutation trinomials over of the form
(1) 
where , and , are two integers, and consequently, four classes of permutation trinomials over with the form (1) were obtained in [12] based on some subtle manipulation of solving equations with low degree over finite fields, and another two classes of such permutations were presented in [13] by virtue of the property of linear fractional polynomials over finite fields. Meanwhile, some similar and more general results on permutation trinomials over were also obtained in [5, 11]. For the permutation polynomials from Niho exponents, the reader is referred to [20, 21, 22] for some recent results and to a survey paper [14]. Very recently, followed the work of [12], by some delicate operation of solving equations with low degrees over finite fields, Deng and Zheng [1] presented two more classes of permutation trinomials over of the form (1), and proposed a conjecture on such a kind of permutation trinomials based on computer experiments. This paper is devoted to settle the conjecture proposed by Deng and Zheng in [1].
2 A conjecture on permutation trinomials of the form (1)
A criterion for a polynomial of the form (1) to be a permutation polynomial had been characterized by the following lemma which was proved by Park and Lee [17], Wang [25] and Zieve [28].
Lemma 1.
(1) ;
(2) permutes the set of the th roots of unity in .
From now on, let be a positive integer and denote the th roots of unity in , i.e., the unit circle of by
Note that if . Then, according to Lemma 1, to prove Conjecture 1, it suffices to show , or equivalently
(2) 
permutes the unit circle of . Observe that is irreducible of which implies that has solutions in if and only if . Thus, by , one gets for any . It is therefore (2) can be written as
(3) 
Then, to prove Conjecture 1, it suffices to show that (3) permutes the unit circle of if , i.e.,
has a unique solution in for any if , which is equivalent to proving that the equation
(4) 
has at most one solution in for any if .
3 Proof of Conjecture 1
This section presents the proof of Conjecture 1.
Lemma 2.
Let , where . If , where , is a factor of , then , must satisfy one of the following conditions:
(1) and ;
(2) and ;
(3) .
Proof.
Assume that can be factorized as
Expanding the right hand side of and comparing the coefficients of where gives
and comparing the coefficients of for gives
Then, according to the values of and , one gets
i.e.,
(5)  
(6) 
In the following we shall consider three cases to prove Lemma 2.
Case 1. If , i.e., since . Then by (5), one obtains that , i.e., . Replacing by gives
Thus, in this case we have
(7)  
(8) 
To prove Conjecture 1, we need to show that in Lemma 2 cannot have two solutions in for any , i.e., cannot have a quadratic factor satisfying have two solutions in . Observe that if are two solutions to , then
Moreover, one has
i.e., . This implies that if is a factor of satisfying has two solutions in , then it must have . Actually this fact has been found in [22] and the number of solutions in to has also been characterized there. We provide the proof of the relation here to make the paper selfcontained.
Due to this fact, we further consider the conditions in Lemma 2.
Lemma 3.
Let , where . If , where and , is a quadratic factor of , then must satisfy
(11) 
where and .
Proof.
According to Lemma 2, we can discuss the three cases in Lemma 2 as follows:

By , one gets , , which implies
Let and , then we have

;

;

;

.

Thus, by Lemma 2 (3), one obtains
This completes the proof. ∎
Notice that if (4) has two or more solutions in for some in , then must have a quadratic factor, say . If so, by Lemma 3, the coefficients , must satisfy the relation (11). Then, to prove Conjecture 1, we only need to show that (11) has no solution in if ,
To this end, define
Notice that is irreducible over and , then all the roots of are conjugate over and lie in [15, Thm. 2.14]. Define
Lemma 4.
The polynomial can be factorized over as follows:
Proof.
Since all the roots of are distinct and lie in , thus to complete the proof, we only need to show that if for any . By a direct computation, if , then one gets
which implies that
Note that . Then, , and , i.e., for any and any . This implies the proof. ∎
Lemma 5.
has no solution if .
Proof.
Suppose that , then taking th power on both sides of (17) gives
(18) 
Notice that . Otherwise we have , a contradiction to the facts that and satisfying . Then, by (17) and (18), one can obtain that
Again by , i.e., , one has
then , which leads to
a contradiction to and . This completes the proof. ∎
4 Conclusion
In this paper, by analyzing the possible quadratic factors of an th degree polynomial over the finite field , a conjecture on permutation trinomials over proposed by Deng and Zheng in [1] was settled.
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