Effective Theorems for Quadratic Spaces over the Algebraic Closure of Q

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Mathematics (CMC)

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Let N ≥ 2 be an integer, F a quadratic form in N variables over Q, and Z ⊆ QN an L-dimensional subspace, 1 ≤ LN. We prove the existence of a small-height maximal totally isotropic subspace of the bilinear space (Z, F). This provides an analogue over Q of well-known theorems of Schlickewei-Schmidt and Vaaler proved respectively over Q and over a number field. We use our result to prove an effective version of Witt orthogonal decomposition for a bilinear space over Q. We also demonstrate an orthogonal version of Siegel’s lemma for a bilinear space over Q. This extends previous results of the author over a number field. All bounds on height are explicit.

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© 2006 Lenny Fukshansky

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