On two exponential sums and their applications
Let \(F_{q}\) be a finite field with \(q=p^{s}\) elements. The paper concerns the problem on existence of primitive polynomials \[ f(x)=x^{n}+a_{1}x^{n-1}+ \cdots +a_{n-1}x+a_{n} \in F_{q}[x] \] with the \(i\)-th coefficient \(a_{i}=a\) prescribed. Let \(N(q,n,i,a)\) denote the number of such polynomials. It was conjectured by \textit{T. Hansen} and \textit{G. Mullen} [Math. Comput. 59, 639-643 (1992; Zbl 0770.11053)] that \(N(q,n,i,a) > 0\) for all \((q,n,i,a)\) with the only three nontrivial exceptions \((q,n,i,a)=(4,3,1,0), (4,3,2,0), (2,4,2,1)\). This is true for \(i=1\) [see \textit{S. D. Cohen}, Lect. Notes Pure Appl. Math. 141, 43-55 (1993; Zbl 0812.11069)], and also for \(i=2\) when \(q\) is odd [\textit{W.-B. Han}, Math. Comput. 65, 331-340 (1996; Zbl 0865.11081)]. If now \(q\) is even and \(n \geq 4\), the author proves that \(N(q,n,2,a) > 0\) for all pairs \((n,a)\) that differ from \((4,0), (5,0)\) and \((6,0)\).
scientific article; zbMATH DE number 1199971
scientific article