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Acta Mathematica Sinica

, Volume 9, Issue 4, pp 382–389 | Cite as

On small zeros of quadratic forms over finite fields (II)

  • Wang Yuan
Article

Abstract

Let\(Q(\underline{\underline x} ) = Q(x_1 , \cdot \cdot \cdot x_n )\) be a quadratic form with integer coefficients and letp denote a prime. Cochrane[1] proved that ifn≥4 then\(Q(\underline{\underline x} ) = 0(\bmod p)\) has a solution\(\underline{\underline x} \ne \underline{\underline 0} \) satisfying\(\left| {\underline{\underline x} } \right| \ll \sqrt p \), where\(\left| {\underline{\underline x} } \right| = \max \left| {x_i } \right|\). The aim of the present paper is to generalize the above result to finite fields.

Keywords

Characteristic Function Quadratic Form Nonnegative Integer Finite Field Fourier Coefficient 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    Todd Cochrane,Small zeros of quadratic forms modulo p, III, J. of Number Theory,37 (1991), 92–99.CrossRefMathSciNetzbMATHGoogle Scholar
  2. [2]
    Heath-Brown, D.R.,Small solutions of quadratic congruences, Glasgov Math. J.,27 (1985), 87–93.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    Wang Yuan,On small zeros of quadratic forms over finite fields, J. of Number Theory,31 (1989), 272–284.CrossRefzbMATHGoogle Scholar

Copyright information

© Science Press 1993

Authors and Affiliations

  • Wang Yuan
    • 1
  1. 1.Institute of MathematicsAcademia SinicaBeijingChina

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