Quadratic forms and their duals

TL;DR

This paper generalizes and unifies existing results on the dualization of quadratic forms and quadrics in projective spaces, especially addressing cases in characteristic two fields.

Contribution

It introduces a framework for dual quadratic forms defined on subspaces, extending previous results and clarifying the relation between forms and their duals in various characteristics.

Findings

Established conditions for dual quadratic forms in characteristic two

Unified various dualization results in the literature

Provided a relation between vectors and linear forms in dual forms

Abstract

There are many specific results, spread over the literature, regarding the dualisation of quadrics in projective spaces and quadratic forms on vector spaces. In the present work we aim at generalising and unifying some of these. We start with a quadratic form $Q$ that is defined on a subspace $S$ of a finite-dimensional vector space $V$ over a field $F$. Whenever $Q$ satisfies a certain condition, which comes into effect only when $F$ is of characteristic two, $Q$ gives rise to a dual quadratic form $\hat{Q}$. The domain of the latter is a particular subspace $\hat{S}$ of the dual vector space of $V$. The connection between $Q$ and $\hat{Q}$ is given by a binary relation between vectors of $S$ and linear forms belonging to $\hat{S}$.