Milner 317

Wednesdays, 12:30-1:30 PM

Texas A&M University

**Wednesday, January 25
Milner 313, 12:30 PM**

**Title:** *Effective theorems for quadratic spaces over Q-bar*

**Abstract:** 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.