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# On Effective Witt Decomposition and the Cartan--Dieudonn{é Theorem

Published:2007-12-01
Printed: Dec 2007
• Lenny Fukshansky
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## Abstract

Let $K$ be a number field, and let $F$ be a symmetric bilinear form in $2N$ variables over $K$. Let $Z$ be a subspace of $K^N$. A classical theorem of Witt states that the bilinear space $(Z,F)$ can be decomposed into an orthogonal sum of hyperbolic planes and singular and anisotropic components. We prove the existence of such a decomposition of small height, where all bounds on height are explicit in terms of heights of $F$ and $Z$. We also prove a special version of Siegel's lemma for a bilinear space, which provides a small-height orthogonal decomposition into one-dimensional subspaces. Finally, we prove an effective version of the Cartan--Dieudonn{\'e} theorem. Namely, we show that every isometry $\sigma$ of a regular bilinear space $(Z,F)$ can be represented as a product of reflections of bounded heights with an explicit bound on heights in terms of heights of $F$, $Z$, and $\sigma$.
 Keywords: quadratic form, heights
 MSC Classifications: 11E12 - Quadratic forms over global rings and fields 15A63 - Quadratic and bilinear forms, inner products [See mainly 11Exx] 11G50 - Heights [See also 14G40, 37P30]

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