Incomplete additive character sums and applications. (English) Zbl 1019.11034

Jungnickel, Dieter (ed.) et al., Finite fields and applications. Proceedings of the fifth international conference on finite fields and applications \(F_q5\), University of Augsburg, Germany, August 2-6, 1999. Berlin: Springer. 462-474 (2001).
Let \(F_{q}\) be a finite field, \(S\) a subset of \(F_{q}\), \( \chi\) a multiplicative character, \( \psi\) the canonical additive character of \(F_{q}\), and \[ T(g,S)= \sum_{x \in S} \psi(g(x)), \qquad W(f,g,S)= \sum_{x \in S} \chi(f(x)) \psi(g(x)) \] incomplete character sums of polynomials \(f,g \in F_{q}[x]\). The author proves that for any \(S \subset F_{q}\) and \(n \geq 2\) with \((n,q-1)\) there exists an irreducible polynomial \(g \in F_{q}[x]\) of degree \(n\) such that \[ |T(g,S)|\geq \sqrt{|S|\left( 1+{|S|-1 \over (q-1)^{2}} \right)}. \] Similarly, for any \(S \subset F_{q}\), \(m \geq 2\) with \(\text{ord} (\chi) \nmid m\), and any \(g \in F_{q}[x]\) there exists a monic irreducible polynomial \(f \in F_{q}[x]\) of degree \(m\) such that \[ |W(f,g,S)|\geq \sqrt{|S|} \] [see also A. Winterhof, Des. Codes Cryptography 22, 123-131 (2001; Zbl 0995.11067)] and S. A. Stepanov [Arithmetic of algebraic curves, Plenum, New York (1994; Zbl 0862.11036), pp. 85-86].
The author gives also a series of new upper bounds for the above character sums. Let \(q=p^{\nu}\) where \(p\) is a prime, \( \{\omega_{0}, \ldots, \omega_{\nu-1} \}\) be a basis of \(F_{q}\) over \(F_{p}\), and \(g \in F_{q}[x]\) a polynomial of degree \(n\) with \((n,q)=1\). If \[ B= \{x=x_{0} \omega_{0}+ \cdots +x_{\nu-1} \omega_{\nu-1} \mid 0 \leq x_{i} < k_{i} \leq p, 0 \leq i \leq \nu-1 \} \] and \[ T(g,B)= \sum_{x \in B} \psi(g(x)), \] the author proves that \[ |T(g,B)|\leq(n-1)q^{1/2}(1+ \log p)^{\nu}. \] This extends the corresponding results of Vinogradov, Polya, Davenport and Lewis for linear polynomials, and a result of Burgess for arbitrary polynomials.
Now if \(i= i_{0}+ i_{1}p+ \cdots + i_{\nu-1}p^{\nu-1}\) is a \(p\)-adic representation of a positive integer \(i\), let \[ S= \{x_{i}= i_{0} \omega_{0}+ \cdots +i_{\nu-1} \omega_{\nu-1} \mid 0 \leq i \leq k-1 \} \] and \[ T(g, S)= \sum_{x_{i} \in S} \psi(g(x_{i})). \] The author proves that \[ |T(g, S)|\leq (n-1)q^{1/2}(1+ \log q). \] Moreover, he shows that \[ |\sum_{x \in A} \psi(g(x))|\leq (n-1)q^{1/2}, \] for any additive subgroup \(A\) of \(F_{q}\). Similar results are obtained for multiplicative character sums \[ U(f,B)= \sum_{x \in B} \chi(f(x)), \quad U(f,S)= \sum_{x_{i} \in S} \chi(f(x_{i})), \quad U(f,A)= \sum_{x \in A} \chi(f(x)) \] and for hybrid sums \[ W(f,g,B)= \sum_{x \in B} \chi(f(x)) \psi(g(x)),\quad W(f,g,S)= \sum_{x_{i} \in S} \chi(f(x_{i})) \psi(g(x_{i})) \] and \[ W(f,g,A)= \sum_{x \in A} \chi(f(x)) \psi(g(x)). \]
The above results are applied to a variant of Waring’s problem in finite fields, the distribution properties of irreducible polynomials in \(F_{q}[x]\), and the study of dc-constrained codes with additive characters.
For the entire collection see [Zbl 0959.00027].


11T23 Exponential sums
11L07 Estimates on exponential sums
94B15 Cyclic codes
11T71 Algebraic coding theory; cryptography (number-theoretic aspects)