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Small zeros of quadratic congruences modulo pq

Published online by Cambridge University Press:  26 February 2010

Todd Cochrane
Todd Cochrane
Affiliation:
Department of Mathematics, Kansas State University, Cardwell Hall, Manhattan, Kansas 66506-2602, U.S.A.
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Let Q(x) = Q(x1, x2,…, xn) be a quadratic form with integer coefficients. Schinzel, Schickewei and Schmidt [9, Theorem 1] have shown that for any modulus m there exists a nonzero such that

and ║x║≤m(1/2)+(1/2(n-1)), where ║x║ = max |xi|. When m is a prime Heath-Brown [8] has obtained a nonzero solution of (1) with ║x║≤m1/2 log m. Yuan [10] has extended Heath-Brown's work to all finite fields. We have proved related results in [5] and [6]. In this paper we extend Heath-Brown's work to moduli which are a product of two primes. Throughout the paper we shall assume that n is even and n>2. For any odd prime p let

where det Q is the determinant of the integer matrix representing Q and is the Legendre symbol.

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Type
Research Article
Information
Mathematika , Volume 37 , Issue 2 , December 1990, pp. 261 - 272
Copyright
Copyright © University College London 1990

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