Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Quadratic geometry of numbers

Authors: Hans Peter Schlickewei and Wolfgang M. Schmidt
Journal: Trans. Amer. Math. Soc. 301 (1987), 679-690
MSC: Primary 11H55
MathSciNet review: 882710
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Abstract: We give upper bounds for zeros of quadratic forms. For example we prove that for any nondegenerate quadratic form $ \mathfrak{F}({x_1}, \ldots ,\,{x_n})$ with rational integer coefficients which vanishes on a $ d$-dimensional rational subspace $ (d > 0)$ there exist sublattices $ {\Gamma _0},\,{\Gamma _1},\, \ldots \,,{\Gamma _{n - d}}$ of $ {\mathbf{Z}^n}$ of rank $ d$, on which $ \mathfrak{F}$ vanishes, with the following properties:

$\displaystyle {\text{rank}}({\Gamma _0} \cap {\Gamma _i}) = d - 1,\quad {\text{rank}}({\Gamma _0} \cup {\Gamma _1} \cup \cdots \cup {\Gamma _{n - d}}) = n$


$\displaystyle {(\det \,{\Gamma _0})^{n - d}}\det \,{\Gamma _1} \cdots \det \,{\Gamma _{n - d}} \ll {F^{{{(n - d)}^2}}}$

, where $ F$ is the maximum modulus of the coefficients of $ \mathfrak{F}$.

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Article copyright: © Copyright 1987 American Mathematical Society