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Siegel’S Lemma Outside Of A Union Of Varieties, Lenny Fukshansky
Siegel’S Lemma Outside Of A Union Of Varieties, Lenny Fukshansky
CMC Faculty Publications and Research
Lecture given at the AMS Special Session on Number Theory, October 2008.
Small Zeros Of Quadratic Forms Over The Algebraic Closure Of Q, Lenny Fukshansky
Small Zeros Of Quadratic Forms Over The Algebraic Closure Of Q, Lenny Fukshansky
CMC Faculty Publications and Research
Let N >= 2 be an integer, F a quadratic form in N variables over (Q) over bar, and Z subset of (Q) over bar (N) an L-dimensional subspace, 1 <= L <= N. We prove the existence of a small-height maximal totally isotropic subspace of the bilinear space (Z, F). This provides an analogue over (Q) over bar of a well-known theorem of Vaaler proved over number fields. We use our result to prove an effective version of Witt decomposition for a bilinear space over (Q) over bar. We also include some related effective results on orthogonal decomposition and structure of isometries for a bilinear space over (Q) over bar. This extends previous results of the author over number fields. All bounds on height are explicit.