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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Small zeros of quadratic forms


Author: Wolfgang M. Schmidt
Journal: Trans. Amer. Math. Soc. 291 (1985), 87-102
MSC: Primary 11E12; Secondary 11E08, 11H50
MathSciNet review: 797047
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Abstract: We give upper and lower bounds for zeros of quadratic forms in the rational, real and $ p$-adic fields. For example, given $ r > 0$, $ s > 0$, there are infinitely many forms $ \mathfrak{F}$ with integer coefficients in $ r + s$ variables of the type $ (r,s)$ (i.e., equivalent over $ {\mathbf{R}}$ to $ X_1^2 + \cdots + X_r^2 - X_{r + 1}^2 - \cdots - X_{r + s}^2$ such that every nontrivial integer zero $ {\mathbf{x}}$ has $ \vert{\mathbf{x}}\vert \gg {F^{r/2s}}$, where $ F$ is the maximum modulus of the coefficients of $ \mathfrak{F}$.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1985-0797047-9
PII: S 0002-9947(1985)0797047-9
Article copyright: © Copyright 1985 American Mathematical Society