Totally Isotropic Subspaces of Small Height in Quadratic Spaces
نویسنده
چکیده
Let K be a global field or Q, F a nonzero quadratic form on KN , N ≥ 2, and V a subspace of KN . We prove the existence of an infinite collection of finite families of small-height maximal totally isotropic subspaces of (V, F ) such that each such family spans V as a K-vector space. This result generalizes and extends a well known theorem of J. Vaaler [16] and further contributes to the effective study of quadratic forms via height in the general spirit of Cassels’ theorem on small zeros of quadratic forms. All bounds on height are explicit.
منابع مشابه
Heights and quadratic forms: Cassels’ theorem and its generalizations
In this survey paper, we discuss the classical Cassels’ theorem on existence of small-height zeros of quadratic forms over Q and its many extensions, to different fields and rings, as well as to more general situations, such as existence of totally isotropic small-height subspaces. We also discuss related recent results on effective structural theorems for quadratic spaces, as well as Cassels’-...
متن کاملEffective Structure Theorems for Symplectic Spaces via Height
Given a 2k-dimensional symplectic space (Z, F ) in N variables, 1 < 2k ≤ N , over a global field K, we prove the existence of a symplectic basis for (Z, F ) of bounded height. This can be viewed as a version of Siegel’s lemma for a symplectic space. As corollaries of our main result, we prove the existence of a small-height decomposition of (Z, F ) into hyperbolic planes, as well as the existen...
متن کاملar X iv : 0 80 1 . 47 73 v 1 [ m at h . N T ] 3 0 Ja n 20 08 EFFECTIVE STRUCTURE THEOREMS FOR SYMPLECTIC SPACES VIA HEIGHT
Given a 2k-dimensional symplectic space (Z, F) in N variables, 1 < 2k ≤ N , over a global field K, we prove the existence of a symplectic basis for (Z, F) of bounded height. This can be viewed as a version of Siegel's lemma for a symplectic space. As corollaries of our main result, we prove the existence of a small-height decomposition of (Z, F) into hyperbolic planes, as well as the existence ...
متن کاملAntidesigns and regularity of partial spreads in dual polar graphs
We give several examples of designs and antidesigns in classical finite polar spaces. These types of subsets of maximal totally isotropic subspaces generalize the dualization of the concepts of m-ovoids and tight sets of points in generalized quadrangles. We also consider regularity of partial spreads and spreads. The techniques that we apply were developed by Delsarte. In some polar spaces of ...
متن کاملLagrangian pairs and Lagrangian orthogonal matroids
Represented Coxeter matroids of types Cn and Dn, that is, symplectic and orthogonal matroids arising from totally isotropic subspaces of symplectic or (even-dimensional) orthogonal spaces, may also be represented in buildings of type Cn and Dn, respectively (see [4, Chapter 7]). Indeed, the particular buildings involved are those arising from the flags or oriflammes, respectively, of totally is...
متن کامل