On Effective Witt Decomposition and the Cartan–Dieudonné Theorem
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 KN. 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 σ 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 σ.
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Fukshansky, Lenny. "On Effective Witt Decomposition and Cartan-Dieudonné Theorem." Canadian Journal of Mathematics 59.6 (2007): 1284-1300.