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On Effective Witt Decomposition and the Cartan–Dieudonné Theorem

Published online by Cambridge University Press:  20 November 2018

Lenny Fukshansky
Lenny Fukshansky*
Affiliation:
Department of Mathematics, Mailstop 3368, Texas A&M University, College Station, TX 77843-3368 email: lenny@math.tamu.edu
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Abstract

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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 $\left( Z,\,F \right)$ 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-Dieudonné theorem. Namely, we show that every isometry $~\sigma$ of a regular bilinear space $\left( Z,\,F \right)$ 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

11E1215A6311G50quadratic formsheights
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 59 , Issue 6 , 01 December 2007 , pp. 1284 - 1300
Copyright
Copyright © Canadian Mathematical Society 2007

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