Isotropy of multiples of Pfister forms and weak isotropy of forms over field extensions

TL;DR

This paper investigates the isotropy properties of Pfister forms and their multiples, providing new bounds and insights into the weak isotropy index and level of quadratic forms over field extensions.

Contribution

It introduces an improved lower bound on the Witt index of multiples of Pfister forms and explores the relationship between weak isotropy index and level of quadratic forms.

Findings

New lower bound on Witt index of Pfister multiples

Characterization of admissible and inadmissible weak isotropy indices

Resolution of open questions on levels of round and Pfister forms

Abstract

The isotropy of multiples of Pfister forms is studied. In particular, an improved lower bound on the value of their first Witt index is obtained. This result and certain of its corollaries are applied to the study of the weak isotropy index (or equivalently, the sublevel) of arbitrary quadratic forms. The relationship between this invariant and the level of the (quadratic) form is investigated. The problem of determining the set of values of the weak isotropy index of a form with respect to field extensions is addressed, with both admissible and inadmissible integers being determined. The analogous investigation with respect to the level of a form is also undertaken, with some questions asked by Berhuy, Grenier-Boley and Mahmoudi being resolved. An examination of the weak isotropy index and the level of round and Pfister forms concludes this article.