Rationality of four-valued families of Weil sums of binomials (Q6556230)

Studying exponential and character sums is an intriguing subject in the theory of finite fields. Daniel J. Katz has authored numerous papers in this area of study. This article is a very recent contribution to the theory of finite fields.\N\NLet \(K\) be a finite field of characteristic \(p\) and order \(q = p^{n}\). Let \(\xi = \exp^{(\frac{2\pi i}{p})}\). The canonical additive character of \(K\) is \(\psi: K \rightarrow \mathbb Q(\xi)\) given by \(\psi(x) = \xi^{Tr(x)}\), where \(Tr: K \rightarrow \mathbb F_{p}\) with \(Tr(x) = x+x^{p}+\cdots+x^{\frac{q}{p}}\). Let \(s\) to denote an invertible exponent over \(K\), that is, a positive integer with \(\gcd(s, q-1) = 1\). This ensures that \(s\) has a multiplicative inverse, \(s^{-1}\), modulo \(q -1\) and makes \(x \mapsto x^{s}\) a permutation of the field K with inverse map \(x \mapsto x^{s^{-1}}\). For each \(u \in K\), we define \[W _{K,s}u = \sum_{x\in K} \psi(x^{s} - ux) = \sum_{x\in K}\xi ^{Tr(x^{s}-ux)}\] which is a Weil sum of a binomial (if \(u \neq 0\)) or a Weil sum of a monomial (if \(u = 0\)). A \textit{multiset} of elements from a set \(X\) is a function \(\mu\) from \(X\) into the nonnegative integers, where for \(x \in X\) the value \(\mu(x)\) is the \textit{frequency }(number of instances) of \(x\) in the multiset. Thus, \(\mu\) represents a normal set if and only if it maps \(X\) into \({0, 1}\) (in which case \(\mu\) is identified with the subset \(\mu ^{-1}({1})\) of X). The \textit{Weil spectrum} for the field \(K\) and the exponent \(s\) is the multiset of values \(W ^{K,s}_{u}\) as \(u\) runs through \(K^{\times}\).\N\NThe main results are the following:\N\NTheorem: Let \(K\) be a finite field of characteristic \(p\) and \(s\) be an invertible exponent over \(K\). Then the Weil spectrum for \(K\) and \(s\) is rational if and only if \(s \equiv 1 \pmod{p - 1}\).\N\NTheorem: Let \(K\) be a finite field and \(s\) be an invertible exponent over \(K\). If the Weil spectrum for \(K\) and \(s\) is \(3-\)valued, then it is rational.\N\NTheorem. Let \(K\) be a finite field and \(s\) be an invertible exponent over \(K\). If the Weil spectrum for \(K\) and \(s\) is \(4-\)valued, then it is rational unless \(K = \mathbb{F}_{5}\) and \(s \equiv 3 \pmod 4\) (in which case \(W_{K,s} = \{\frac{(5 \pm\sqrt{ 5})}{2}, \pm \sqrt{5}\})\).\N\NThis article unveils a promising avenue for furthering the theory of finite fields, presenting an opportunity to generate enhanced outcomes in higher valued families of Weil sums of binomials.