On Effective Witt Decomposition and Cartan-dieudonné Theorem
نویسنده
چکیده
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 . A classical theorem of Witt states that the bilinear space (Z,F ) can be decomposed into an orthogonal sum of hyperbolic planes, 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 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 small heights with an explicit bound on heights in terms of heights of F , Z, and σ.
منابع مشابه
Quadratic Forms Chapter I: Witt’s Theory
1. Four equivalent definitions of a quadratic form 2 2. Action of Mn(K) on n-ary quadratic forms 4 3. The category of quadratic spaces 7 4. Orthogonality in quadratic spaces 9 5. Diagonalizability of Quadratic Forms 11 6. Isotropic and hyperbolic spaces 13 7. Witt’s theorems: statements and consequences 16 7.1. The Chain Equivalence Theorem 18 8. Orthogonal Groups 19 8.1. The orthogonal group o...
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