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

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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.

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References

  1. Todd Cochrane,Small zeros of quadratic forms modulo p, III, J. of Number Theory,37 (1991), 92–99.

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  2. Heath-Brown, D.R.,Small solutions of quadratic congruences, Glasgov Math. J.,27 (1985), 87–93.

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  3. Wang Yuan,On small zeros of quadratic forms over finite fields, J. of Number Theory,31 (1989), 272–284.

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Yuan, W. On small zeros of quadratic forms over finite fields (II). Acta Mathematica Sinica 9, 382–389 (1993). https://doi.org/10.1007/BF02560131

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