Acta Mathematica Sinica

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

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

  • Wang Yuan


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.


Characteristic Function Quadratic Form Nonnegative Integer Finite Field Fourier Coefficient 
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  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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