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Exact values of the sums of multiplicative characters of polynomials over finite fields. (English. Russian original) Zbl 1273.11173

Math. Notes 88, No. 3, 308-316 (2010); translation from Mat. Zametki 88, No. 3, 340-349 (2010).
Let \(p\) be a prime number, \(q=p^{\nu}\) a power of \(p\), \(\mathbb F_q\) a finite field with \(q\) elements and \(\mathbb F_{q^m}\) a finite extension of \(\mathbb F_q\) of degree \(m \geq 1\). If \( \beta\) is a fixed primitive element of \(\mathbb F_{q^m}\) and \(s \geq 1\) is a divisor of \(q^{m}-1\), let \[ \chi_{s,m}( \beta^{u})= \exp \left\{ \frac{2 \pi i}{s} \cdot u \right\}, \quad u=0,1, \ldots ,q^{m}-2, \] denote a fixed multiplicative character of the field \(\mathbb F_{q^m}\) of order \(s\). The authors prove the following two results:
(i) Let \(f(x)\) be a polynomial from \(\mathbb F_{q^m}[x]\). Then \[ \sum_{x \in \mathbb F_{q^m}} \chi_{s,m}(f(x))=q^{m}- \# \{x \in \mathbb F_{q^m} \mid f(x)=0 \} \] if and only if there exists a polynomial \(g(x) \in\mathbb F_{q^m}[x]\) such that \[ f(x)=(g(x))^{s} \pmod{x^{q^m}-x}; \] (ii) Let \(l \geq 2\) be an integer such that \(l\mid m\) and \(l\mid s\). If \(f(x) \in\mathbb F_{q^m}[x]\) is a polynomial such that all of its values for \(x \in\mathbb F_{q^m}\) lie in the subfield \(\mathbb F_{q^{m/l}}\) then \[ \sum_{x \in \mathbb F_{q^m}} \chi_{s,m}(f(x)^{s/l}) = q^m- \# \{x \in\mathbb F_{q^m} \mid f(x)=0 \}. \]
This generalizes the results obtained earlier by the reviewer [Discrete Math. Appl. 2, No. 5, 523–532 (1992); translation from Diskretn. Mat. 3, No. 2, 77–86 (1991; Zbl 0736.11070)] and M. M. Glukhov [Discrete Math. Appl. 4, No. 5, 467–472 (1994); translation from Diskretn. Mat. 6, No. 3, 136–142 (1994; Zbl 0832.11043)]. The final section of paper concerns the special case of prime finite fields and character sums of polynomials with the quadratic character (Legendre symbol). In this case the authors improve the corresponding results of A. A. Karatsuba [Mat. Zametki 14, 67–72 (1973; Zbl 0277.10029)], D. A. Mit’kin [Usp. Mat. Nauk 30, No. 5(185), 214 (1975; Zbl 0314.10025)], A. Tietäväinen [J. Lond. Math. Soc. (2) 29, 204–210 (1984; Zbl 0499.10039)], H. Tarnanen [Discrete Math. 57, 285–295 (1985; Zbl 0592.10032)] and the reviewer [Proc. Steklov Inst. Math. 143, 187–189 (1980); translation from Tr. Mat. Inst. Steklova 143, 175–177 (1977; Zbl 0431.10023)].

MSC:

11T24 Other character sums and Gauss sums
11G20 Curves over finite and local fields
33E50 Special functions in characteristic \(p\) (gamma functions, etc.)
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References:

[1] R. Lidl and H. Niederreiter, Finite Fields (Addison-Wesley, Reading, Mass., 1983; Mir, Moscow, 1988), Vol. 1, 2.
[2] S. A. Stepanov, ”On lower bounds for character sums over finite fields,” Diskretn. Mat. 3(2), 77–86 (1991) [Discrete Math. Appl. 2 (5), 523–532 (1991)]. · Zbl 0736.11070
[3] M. M. Glukhov, ”Lower bounds for character sums over finite fields,” Diskretn. Mat. 6(3), 136–142 (1994) [Discrete Math. Appl. 4 (5), 467–472 (1994)]. · Zbl 0832.11043
[4] A. A. Karatsuba, ”Lower bounds for character sums of polynomials,” Mat. Zametki 14(1), 67–72 (1973).
[5] D. A. Mit’kin, ”Lower bounds for the sums of Legendre symbols and trigonometric sums,” Uspekhi Mat. Nauk 30(5), 214 (1975).
[6] S. A. Stepanov, ”On lower estimates of incomplete character sums of polynomials,” in Trudy Mat. Inst. Steklov, Vol. 143: Analytic Theory of Numbers, Mathematical Analysis, and Their Applications, Collection of papers, Dedicated to the 85th anniversary of Academician I. M. Vinogradov (Nauka, Moscow, 1977), pp. 175–177 [Proc. Steklov Inst. Math. 143, 187–189 (1980)].
[7] V. I. Levenshtein, ”Bounds for packings of metric spaces and some of their applications,” Probl. Kibern. 40, 43–110 (1983).
[8] A. Tietäväinen, ”Lower bounds for the maximum moduli of certain character sums,” J. London Math. Soc. (2) 29(2), 204–210 (1984). · Zbl 0499.10039 · doi:10.1112/jlms/s2-29.2.204
[9] H. Tarnanen, ”On character sums and codes,” Discrete Math. 57(3), 285–295 (1985). · Zbl 0592.10032 · doi:10.1016/0012-365X(85)90180-3
[10] A. A. Karatsuba, ”Arithmetic problems in the theory of Dirichlet characters,” Uspekhi Mat. Nauk 63(4), 43–92 (2008) [Russian Math. Surveys 63 (4), 641–690 (2008)]. · doi:10.4213/rm9234
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