Let and be any positive integers with . Let and where is a prime. We consider the polynomial
Notice the more general polynomial forms with and can be transformed into this form by the substitution . It is clear that have no multiple roots.
These polynomials have arisen in several different contexts including finite geometry, the inverse Galois problem , the construction of difference sets with Singer parameters , determining cross-correlation between -sequences [11, 14] and to construct error-correcting codes . These polynomials are also exploited to speed up (the relation generation phase in) the index calculus method for computation of discrete logarithms on finite fields [12, 13] and on algebraic curves .
Let denote the number of zeros in of polynomial and denote the number of such that has exactly zeros in . In 2004, Bluher  proved that takes either of 0, 1, 2 and where and computed for every . She also stated some criteria for the number of the -zeros of .
The ultimate goal in this direction of research is to identify all the -zeros of . Subsequently, there were much efforts for this goal, specifically for a particular instance of the problem over binary fields i.e. . In 2008 and 2010, Helleseth and Kholosha [16, 17] found new criteria for the number of -zeros of . In the cases when there is a unique zero or exactly two zeros and
is odd, they provided explicit expressions of these zeros as polynomials of. In 2014, Bracken, Tan and Tan  presented a criterion for in when and is even. Very recently, Kim and Mesnager  completely solved this equation over when . They showed that the problem of finding zeros in of in fact can be divided into two problems with odd : to find the unique preimage of an element in under a MCM polynomial and to find preimages of an element in under a Dickson polynomial. By completely solving these two independent problems, they explicitly calculated all possible zeros in of , with new criteria for which is equal to , or as a by-product.
Very recently, new criteria for which has , , or roots were stated by  for any characteristic.
We discuss the equation , without any restriction on and . After defining a sequence of polynomials and considering its properties in Section 2, it is shown in Section 3 that if then there exists a quadratic equation that the rational zeros must satisfy. In Section 4, we state some useful properties of the polynomials which appear as the coefficients of that quadratic equation. In Section 5, new criteria for the number of the -zeros of are proved. For the cases of one or two -zeros, we provide explicit expressions for these rational zeros in terms of . We also provide a parametrization of the ’s for which has rational zeros. Based that parametrization, all the rational zeros are also expressed. For the case of rational zeros, some results to explicitly express these rational zeros in terms of are further presented in Section 6. Finally, we conclude in Section 7.
Given positive integers and , define a polynomial
Usually we will abbreviate as . For , is the absolute trace of . The zeros of this polynomial are studied in [KCLGM2019]. In particular, we need the following.
For any positive integers and ,
Evidently, . The linear mapping has the kernel and so . On the other hand, can not have a kernel of greater cardinality than its degree .∎
Define the sequence of polynomials as follows:
The following lemma gives another identity which can be used as an alternative definition of and an interesting property of this polynomial sequence which will be importantly applied afterwards.
For any , the following are true.
The zero set of can be completely determined for all :
For any ,
Proof. Given any , there exists at least one element such that and . Then, for any , we have
where for it is assumed that the product over the empty set is equal to 1. Indeed, this can be proved by induction on as follows. For and , we have
Assuming this identity holds for all indices less than , we have
Thus if and only if and , which by Proposition 1 is equivalent to for some .
Therefore, if and only if for some . ∎
3 Quadratic equation satisfied by rational zeros of
Letting , define polynomials
We will show that if then the -zeros of satisfy a quadratic equation and therefore necessarily .
Let . If for then
If for , then and thus we get
Now, we prove that for any
This shows that (6) holds for and so for all .
Taking in (6) and using the fact that when , we obtain the result of the lemma.∎
4 Some equalities involving and
To determine the rational zeros of when , we will need the following properties of the polynomials and which appear as coefficients of the quadratic equation (4).
For any , the following are true.
The first item follows from
The second item is proved as follows. Let . Then
Consider . By substituting this and using (1), we have
By the way, since
we get that is,
Finally, the third item is verified as follows:
For even, we will further need the following proposition.
Let . Let with . Let and . The followings hold.
5 Rational zeros of
By exploiting the results of previous sections, now we can completely solve the equation in arbitrary finite fields.
Let . The following are equivalent.
i.e. has exactly zeros in .
, or equivalently by Proposition 2, there exists such that .
There exists such that . Then the zeros in of are and for .
We already showed that if , then i.e. .
If i.e. there exists such that , then the set given by
is the set of all zeros of . In fact, the cardinality of this set is exactly as is not in . Also we have
Thus splits in . Corollary 7.2 of  states that splits in if and only if has exactly zeros in .
To begin with, define , , and .
Now, we will show that the mapping
which is well-defined by Proposition 2 and the equivalence between Item 1 and Item 2, is surjective.
Regarding , we can write where , ,