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Some estimates for character sums and applications. (English) Zbl 0995.11067

Let \(p\) be a prime number, \(F_{q}\) a finite field with \(q=p^{ \nu}\) elements, \(S\) a subset of \(F_{q}\), and \( \chi\) a nontrivial multiplicative character of the field \(F_{q}\) of order \(s \geq 2\). If \(n \geq 2\) is an arbitrary integer satisfying \(n \not\equiv 0\pmod s\), the author proves that there exists a monic irreducible polynomial \(f \in F_{q}[x]\) of degree \(n\) such that \[ \left|\sum_{x \in S} \chi(f(x)) \right|\geq \sqrt{|S|}. \] This extends the author’s result [Finite Fields Appl. 4, 43-54 (1998; Zbl 0910.11055)] concerning the case \(n=1\).
Let \( \omega_{1}, \ldots , \omega_{ \nu}\) be a basis of \(F_{q}\) over its prime subfield \(F_{p}= \{0,1, \ldots ,p-1 \}\), and \[ S= \{x=x_{1} \omega_{1}+ \cdots +x_{ \nu} \omega_{ \nu} \in F_{q} \mid 0\leq x_{i} \;\leq H_{i}, 1 \leq i \leq \nu \}, \] for any \(0 \leq H_{i} < p\). If \(f \in F_{q}[x]\) is a polynomial of degree \(n \geq 1\) that is not an \(s\)-th power and has \(m\) distinct roots in its splitting field over \(F_{q}\), the author shows that \[ \left|\sum_{x \in S} \chi(f(x)) \right|< mq^{1/2}(1+ \log p)^{ \nu}. \] This extends a result of D. A. Burgess [Proc. Lond. Math. Soc. (3) 13, 537-548 (1963; Zbl 0118.04704)] concerning the case \(q=p\) (for some subset \(S\) of a very special form, the author finds a slightly stronger upper bound).
Finally, the author considers some coding theoretic and also other applications of the above results.

MSC:

11T24 Other character sums and Gauss sums
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