Geodesic
On Effective Witt Decomposition and the Cartan–Dieudonné Theorem
Canadian journal of mathematics, Tome 59 (2007) no. 6, pp. 1284-1300

Voir la notice de l'article provenant de la source Cambridge University Press

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 $~\sigma$ .
DOI : 10.4153/CJM-2007-055-4
Mots-clés : 11E12, 15A63, 11G50, quadratic forms, heights 
Fukshansky, Lenny. On Effective Witt Decomposition and the Cartan–Dieudonné Theorem. Canadian journal of mathematics, Tome 59 (2007) no. 6, pp. 1284-1300. doi: 10.4153/CJM-2007-055-4
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