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Small zeros of quadratic forms over number fields. II


Author: Jeffrey D. Vaaler
Journal: Trans. Amer. Math. Soc. 313 (1989), 671-686
MSC: Primary 11E12; Secondary 11H55
DOI: https://doi.org/10.1090/S0002-9947-1989-0940914-7
MathSciNet review: 940914
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Abstract: Let $ F$ be a nontrivial quadratic form in $ N$ variables with coefficients in a number field $ k$ and let $ \mathcal{Z}$ be a subspace of $ {k^N}$ of dimension $ M,1 \leq M \leq N$. If $ F$ restricted to $ \mathcal{Z}$ vanishes on a subspace of dimension $ L,1 \leq L < M$, and if the rank of $ F$ restricted to $ \mathcal{Z}$ is greater than $ M - L$, then we show that $ F$ must vanish on $ M - L + 1$ distinct subspaces $ {\mathcal{X}_0},{\mathcal{X}_1}, \ldots ,{\mathcal{X}_{M - L}}$ in $ \mathcal{Z}$ each of which has dimension $ L$. Moreover, we show that for each pair $ {\mathcal{X}_0},{\mathcal{X}_1},1 \leq l \leq M - L$, the product of their heights $ H({\mathcal{X}_0})H({\mathcal{X}_1})$ is relatively small. Our results generalize recent work of Schlickewei and Schmidt.


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  • [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] -, Addendum to the paper. Bounds for the least solution of homogeneous quadratic equations, Proc. Cambridge Philos. Soc. 52 (1956), 604. MR 0081306 (18:380c)
  • [5] J. H. H. Chalk, Linearly independent zeros of quadratic forms over number fields, Monatsh. Math. 90 (1980), 13-25. MR 593828 (82c:10041)
  • [6] H. Davenport, Note on a theorem of Cassels, Proc. Cambridge Philos. Soc. 53 (1957), 539-540. MR 0086105 (19:125c)
  • [7] -, Homogeneous quadratic equations, Mathematika 18 (1971), 1-4. MR 0292760 (45:1842)
  • [8] S. Raghaven, Bounds of minimal solutions of diophantine equations, Nachr. Akad. Wiss. Gottingen Math. Phys. Kl. 9 (1975), 109-114. MR 0485681 (58:5504)
  • [9] H. P. Schlickewei, Kleine Nullstellen homogener quadratischer Gleichungen, Monatsh. Math. 100 (1985), 35-45. MR 807296 (87f:11019)
  • [10] H. P. Schlickewei and W. M. Schmidt, Quadartic geometry of numbers, Trans. Amer. Math. Soc. 301 (1987), 679-690. MR 882710 (88g:11036)
  • [11] W. M. Schmidt, On heights of algebraic subspaces and diophantine approximations, Ann. of Math. 85 (1967), 430-472. MR 0213301 (35:4165)
  • [12] -, Small zeros of quadratic forms, Trans. Amer. Math. Soc. 291 (1985), 87-102. MR 797047 (86j:11035)
  • [13] R. Schulze-Pillot, Small linearly independent zeros of quadratic forms, Monatsh. Math. 95 (1983), 241-249. MR 712424 (85e:11044)
  • [14] J. D. Vaaler, Small zeros of quadratic forms over number fields, Trans. Amer. Math. Soc. 302 (1987), 281-296. MR 887510 (88j:11018)
  • [15] G. L. Watson, Least solution of homogeneous quadratic equations, Proc. Cambridge Philos. Soc. 53 (1956), 541-543. MR 0086077 (19:120c)

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DOI: https://doi.org/10.1090/S0002-9947-1989-0940914-7
Article copyright: © Copyright 1989 American Mathematical Society

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