On polynomials of special form over a finite field of odd characteristic attaining the Weil bound (Q2577339)

Let \(q=p^l\) and \(Q=q^m\), where \(p\) is a prime, \(l\) and \(m\) are positive integers, \(m\geq 2\), and let \(\mathbb{F}_p\subseteq \mathbb{F}_q\subseteq \mathbb{F}_Q\) be the finite fields of order \(p\), \(q\), and \(Q\), respectively. In [Russ. Math. Surv. 52, No. 2, 271--284 (1997); translation from Usp. Mat. Nauk 52, No. 2, 31--44 (1997; Zbl 0928.11052)] the authors described a class of polynomials \(f(x)\in \mathbb{F}_Q[x]\) of the form \[ f(x)=\sum_{s=0}^{\lfloor m/2\rfloor} a_sx^{1+q^s} \] with nonconstant trace, for which the absolute value of the trigonometric sum \[ S(f)=\sum_{x\in \mathbb{F}_Q} \exp(2\pi i Tr(f(x))/p) \] takes the maximum value. For many examples Weil's bound \[ | S(f)| \leq (\deg(f)-1)Q^{1/2} \] is attained. In [Math. Notes 72, No. 2, 152--157 (2002); translation from Mat. Zametki 72, No. 2, 171--177 (2002; Zbl 1043.11080)] the authors continued their study of \(S(f)\) for the case \(p=2\). In both papers they reduce \(S(f)\) to sums of quadratic forms. In the paper under review they continue their study for the case of odd \(p\). The construction of polynomials for which the Weil bound is attained is based on the construction of cyclic matrices of a given rank. Dickson polynomials of the second kind play an essential role in the study of such matrices.