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Small independent zeros of quadratic forms

Published online by Cambridge University Press:  24 October 2008

R. J. Cook  and
S. Raghavan
R. J. Cook
Affiliation:
Department of Pure Mathematics, University of Sheffield
S. Raghavan
Affiliation:
Tata Institute for Fundamental Research, Bombay, India
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Extract

Let

be a non-degenerate quadratic form with integral coefficients. Further, let Q(x) be a zero form, i.e. let there exist x ≠ 0 in ℤn such that Q(x) = 0. Then we know from Cassels[2], (Davenport[6] and ‘a slightly more general result’ from Birch and Davenport [1]) that there exists a ‘small’ solution x in ℤn of the equation Q(x) = 0; more precisely, if

then there exists x ≠ 0 in ℤn such that Q(x) = 0 and further

(Here, and throughout this section, k will denote a number, not necessarily the same at each occurrence, which depends only on n.) An analogue of this estimate for ‘integral’ quadratic forms over algebraic number fields was proved in [8], with the exponent (n − 1)/2 remaining intact.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 102 , Issue 1 , July 1987 , pp. 5 - 16
Copyright
Copyright © Cambridge Philosophical Society 1987

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References

REFERENCES

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