Small zeros of quadratic forms over number fields
Author:
Jeffrey D. Vaaler
Journal:
Trans. Amer. Math. Soc. 302 (1987), 281-296
MSC:
Primary 11E12
DOI:
https://doi.org/10.1090/S0002-9947-1987-0887510-6
MathSciNet review:
887510
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: Let be a nontrivial quadratic form in
variables with coefficients in a number field
and let
be a
matrix over
. We show that if the simultaneous equations
and
hold on a subspace
of dimension
and
is maximal, then such a subspace
can be found with the height of
relatively small. In particular, the height of
can be explicitly bounded by an expression depending on the height of
and the height of
. We use methods from geometry of numbers over adèle spaces and local to global techniques which generalize recent work of H. P. Schlickewei.
- [1] B. J. Birch and H. Davenport, Quadratic equations in several variables, Proc. Cambridge Philos. Soc. 54 (1958), 135-138. MR 0097355 (20:3824)
- [2] E. Bombieri and J. Vaaler, On Siegel's lemma, Invent. Math. 73 (1983), 11-32. MR 707346 (85g:11049a)
- [3] J. W. S. Cassels, Bounds for the least solution of homogeneous quadratic equations, Proc. Cambridge Philos. Soc. 51 (1955), 262-264. MR 0069217 (16:1002c)
- [4] J. H. H. Chalk, Linearly independent zeros of quadratic forms over number fields, Monatsh. Math. 90 (1980), 13-25. MR 593828 (82c:10041)
- [5] P. Gordan, Über den grössten gemeinsamen Factor, Math. Ann. 7 (1873), 443-448.
- [6] W. V. D. Hodge and D. Pedoe, Methods of algebraic geometry, vol. 1, Cambridge Univ. Press, 1968.
- [7] S. Raghavan, Bounds of minimal solutions of diophantine equations, Nachr. Akad. Wiss. Gottingen, Math. Phys. Kl. 9 (1975), 109-114. MR 0485681 (58:5504)
- [8] H. P. Schlickewei, Kleine Nullstellen homogener quadratischer Gleichungen, Monatsh. Math. 100 (1985), 35-45. MR 807296 (87f:11019)
Retrieve articles in Transactions of the American Mathematical Society with MSC: 11E12
Retrieve articles in all journals with MSC: 11E12
Additional Information
DOI:
https://doi.org/10.1090/S0002-9947-1987-0887510-6
Article copyright:
© Copyright 1987
American Mathematical Society