Effective Theorems for Quadratic Spaces via Height

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Mathematics (CMC)

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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.


This lecture was given during the Number Theory Seminar at UCLA in October 2007, and during the Seminaire de theorie des nombres de Chevaleret at the Institut de Mathématiques de Jussieu, Paris, France in July 2007.

This lecture is also related to the author's lecture titled "Effective structure theorems for quadratic spaces via height," given during the Second International Conference on The Algebraic and Arithmetic Theory of Quadratic Forms 2007, Lake Llanquihue, Chile in December 2007.

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© 2007 Lenny Fukshansky

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