On Effective Witt Decomposition and Cartan-Dieudonné Theorem

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Mathematics (CMC)

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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-Dieudonné theorem on representation of an isometry of a regular symmetrice bilinear space as a product of reflections. Finally, if time permits, I will show a special version of Siegel's Lemma for a bilinear space, which provides a small-height orthogonal decomposition into one-dimensional subspaces.


This lecture was given during the Number Theory Seminar at Texas A&M University in September 2004, and during the West Coast Number Theory Conference in Las Vegas, NV, December 2004.

This lecture is related to an article by the author: "On Effective Witt Decomposition and Cartan-Dieudonné Theorem," from the Canadian Journal of Mathematics Volume 59, Number 6, pages 1284-1300.

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© 2004 Lenny Fukshansky

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