Some New Results Concerning Isotropy of Low-dimensional Forms

Abstract

Let ø and ψ be quadratic forms over a field F of characteristic ≠2. We give an (almost) complete classification of pairs ø, ψ of dimension ≤ 9 such that ø is stably equivalent to ψ. We also study the question when the form ø is isotropic over the function field of ψ. In the case where dim #x00F8; = 9 and dim ψ ≥ 9 we solve this problem completely.

The current draft contains only a list of results. We are planning to write three articles with the following titles:

• (a) Isotropy of 7-dimensional forms and 8-dimensional forms.

• (b) Stable equivalence of 9-dimensional forms.

• (c) Isotropy of 10- and 12-dimensional forms.