Some model theory of quadratic geometries

TL;DR

This paper explores the model theory of quadratic geometries over fields of characteristic two, focusing on their model companions, classifications, and connections to Lie coordinatizable structures.

Contribution

It describes the model companions of orthogonal spaces and quadratic geometries, classifies their pseudo-finite completions, and provides a neostability-theoretic analysis.

Findings

Identification of model companions for orthogonal spaces and quadratic geometries.

Classification of pseudo-finite completions of these theories.

Neostability-theoretic analysis of the model companions.

Abstract

Orthogonal spaces are vector spaces together with a quadratic form whose associated bilinear form is non-degenerate. Over fields of characteristic two, there are many quadratic forms associated to a given bilinear form and quadratic geometries are structures that encode a vector space over a field of characteristic 2 with a non-degenerate bilinear form together with a space of associated quadratic forms. These structures over finite fields of characteristic 2 form an important part of the basic geometries that appear in the Lie coordinatizable structures of Cherlin and Hrushovski. We (a) describe the respective model companions of the theory of orthogonal spaces and the theory of quadratic geometries and (b) classify the pseudo-finite completions of these theories. We also (c) give a neostability-theoretic classification of the model companions and these pseudo-finite completions. This is a small step towards understanding the analogue of the Cherlin-Hrushovski theory of Lie coordinatizable structures in a setting where the involved fields may be pseudo-finite.