Effective Theorems for Quadratic Spaces over the Algebraic Closure of Q
Let N ≥ 2 be an integer, F a quadratic form in N variables over Q, and Z ⊆ QN 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 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.
© 2006 Lenny Fukshansky
Fukshansky, Lenny. "Effective Theorems for Quadratic Spaces over the Algebraic Closure of Q." AMS Special Session on Mahler Measure and Heights, Joint Mathematics Meeting, San Antonio, TX. January 2006.