On Effective Witt Decomposition and the Cartan–Dieudonné Theorem

Abstract Let K be a number field, and let F be a symmetric bilinear form in 2 N 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–Dieudonné 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.