Effective Structure Theorems for Quadratic Spaces and Their Isometries

A classical theorem of Witt states that a bilinear space can be decomposed into an orthogonal sum of hyperbolic planes, singular, and anisotropic components. I will discuss the existence of such a decomposition of bounded height for a symmetric bilinear space over a number field, where all bounds on height are explicit. I will also talk about an effective version of Cartan-Dieudonne theorem on representation of an isometry of a regular symmetric bilinear space as a product of reflections. Finally, if time permits, I will show a special version of Siegel\u27s Lemma for a bilinear space, which provides a small-height orthogonal decomposition into one-dimensional subspaces