On polynomials of special form over a finite field of odd characteristic attaining the Weil bound.

*(English. Russian original)*Zbl 1127.11080
Math. Notes 78, No. 1, 14-22 (2005); translation from Mat. Zametki 78, No. 1, 16-25 (2005).

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.

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.

Reviewer: Arne Winterhof (Linz)

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\textit{L. A. Bassalygo} and \textit{V. A. Zinov'ev}, Math. Notes 78, No. 1, 14--22 (2005; Zbl 1127.11080); translation from Mat. Zametki 78, No. 1, 16--25 (2005)

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##### References:

[1] | R. Lidl and H. Niederreiter, Finite Fields, Encyclopedia of Mathematics and its Applications, vol. 20, Addison-Wesley, Reading, Mass., 1983. · Zbl 0554.12010 |

[2] | L. A. Bassalygo and V. A. Zinov’ev, ”Polynomials of special form over a finite field with maximal absolute value of the trigonometric sum” Uspekhi Mat. Nauk [Russian Math. Surveys], 52 (1997), no. 2, 31–44. |

[3] | A. V. Malyshev, On the Representation of Integers by Positive Quadratic Forms [in Russian], Trudy Mat. Inst. Steklov [Proc. Steklov Inst. Math.], vol. 65, Nauka, Moscow, 1962. |

[4] | L. A. Bassalygo and V. A. Zinov’ev, ”On polynomials over a finite field of even characteristic with maximum absolute value of the trigonometric sum,” Mat. Zametki [Math. Notes], 72 (2002), no. 2, 171–177. · Zbl 1043.11080 |

[5] | R. Lidl and G. L. Mullen, and G. Turnwald, Dickson Polynomials, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 65, John Wiley and Sons, New York, 1993. |

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