Effective Theorems for Quadratic Spaces via Height
A classical theorem of Cassels (1955) asserts that if an integral quadratic form is isotropic over rationals, then it has a non-trivial rational zero of small height, with an explicit bound on height provided. This result has been generalized to number fields by S. Raghavan (1975), and later extended by Schmidt, Schlickewei (1985), and Vaaler (1987) to a statement about existence of small-height maximal totally isotropic subspaces of a quadratic space over a number field, again with explicit bounds on height. I will discuss these results, as well as some of my recent work on an analogue of Schlickewei-Schmidt-Vaaler theorem over Q-bar. I will also talk about the application of these results to an effective version of Witt decomposition for a quadratic space over a number field and over Q-bar. Finally, if time allows, I will also discuss an effective version of the classical Cartan-Dieudonne theorem for a quadratic space.
© 2007 Lenny Fukshansky
Fukshansky, Lenny. "Effective Theorems for Quadratic Spaces via Height." Number Theory Seminar, UCLA. October 2007.