## 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

### Citations:

Zbl 0910.11055; Zbl 0118.04704
Full Text: