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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 quadratic form, heights
MSC Classifications: 11E12, 15A63, 11G50 show english descriptions Quadratic forms over global rings and fields
Quadratic and bilinear forms, inner products [See mainly 11Exx]
Heights [See also 14G40, 37P30]
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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