Effective Theorems for Quadratic Spaces Over Q-bar
Let N >=2 be an integer, F a quadratic form in N variables over Qbar, and Z contained in Qbar^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 Qbar 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 Qbar. If time allows, we will also discuss some related effective results on orthogonal decomposition and structure of isometries for a bilinear space over Qbar. This extends previous results of the author over number fields. All bounds on height are explicit.
© 2006 Lenny Fukshansky
Fukshansky, Lenny. "Effective Theorems for Quadratic Spaces Over Q-bar." Number Theory Seminar, Texas A&M University, College Station, Texas. 25 January 2006.