zbMATH — the first resource for mathematics

On polynomials over a finite field of even characteristic with maximum absolute value of the trigonometric sum. (English. Russian original) Zbl 1043.11080
Math. Notes 72, No. 2, 152-157 (2002); translation from Mat. Zametki 72, No. 2, 171-177 (2002).
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 \[ S(f)=\sum_{x\in \mathbb{F}_Q} \exp(2\pi i \text{Tr}(f(x))/p) \] takes the maximum value. For many examples Weil’s bound \[ | S(f)| \leq (\deg(f)-1)Q^{1/2} \] is attained. The result is obtained by reducing \(S(f)\) to trigonometric sums of a quadratic form \(z_1z_2\). In the paper under review the study of the case \(p=2\) and \(m\geq 3\) is continued by reducing \(S(f)\) to sums of quadratic forms \(z_1z_2+z_3z_4\). If \(m=2k\), \(k\geq 4\), the authors construct at least \((q^k-q^3)/(2(q-1))\) such polynomials of degree \(q^{k-2}+1\). If \(m=2k+1\), \(k\geq 1\), they show that the maximal possible value of \(| S(f)| \) is \(q^{2k-1}\).

11T23 Exponential sums
11T06 Polynomials over finite fields
Full Text: DOI