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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

[1] Birch, B. J. and Davenport, H.. Quadratic equations in several variables. Proc. Cambridge Philos. Soc. 54 (1958), 135–138.CrossRefGoogle Scholar
[2] Cassels, J. W. S.. Bounds for the least solutions of homogeneous quadratic equations. Proc. Cambridge Philos. Soc. 51 (1955), 262–264.CrossRefGoogle Scholar
[3] Cassels, J. W. S.. Addendum to Bounds for…equations. Proc. Cambridge Philos. Soc. 52 (1956), 604.CrossRefGoogle Scholar
[4] Cassels, J. W. S.. Rational quadratic forms (Academic Press, 1978).Google Scholar
[5] Chalk, J. H. H.. Linearly independent zeros of quadratic forms over algebraic number-fields. Monatsh. Math. 90 (1980), 13–25.CrossRefGoogle Scholar
[6] Davenport, H.. Note on a theorem of Cassels. Proc. Cambridge Philos. Soc. 53 (1957), 539–540.CrossRefGoogle Scholar
[7] Davenport, H.. Homogeneous quadratic equations. Mathematika 18 (1971), 1–4.CrossRefGoogle Scholar
[8] Raghavan, S.. Bounds for minimal solutions of diophantine equations. Nachr. Akad. Wiss. Gottingen, Math-Phys. Kl. No. 9 (1975), 109–114.Google Scholar
[9] Schlickewei, H. P.. Kleine Nullstellen homogener quadratischer Gleichungen. Monatsh. Math. 100 (1985), 35–46.CrossRefGoogle Scholar
[10] Schmidt, W. M.. Small zeros of quadratic forms (preprint).Google Scholar
[11] Schulze-Pillot, R.. Small linearly independent zeros of quadratic forms. Monatsh. Math. 95 (1983), 241–249.CrossRefGoogle Scholar
[12] Watson, G. L.. Least solutions of homogeneous quadratic equations. Proc. Cambridge Philos. Soc. 53 (1957), 541–543.CrossRefGoogle Scholar