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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 $$q^{2k-1}$$.

##### MSC:
 11T23 Exponential sums 11T06 Polynomials over finite fields
##### Keywords:
finite fields; trigonometric sums; Weil’s bound; polynomials
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