Linkage of Quadratic Pfister Forms

TL;DR

This paper investigates the conditions under which sets of quadratic n-fold Pfister forms share a common (n-1)-fold factor, introducing an invariant in higher powers of the fundamental ideal to characterize such relationships.

Contribution

It introduces a new invariant in the Witt ring that detects when a set of quadratic Pfister forms share a common factor, expanding understanding of their algebraic structure.

Findings

The invariant lives in I_q^{n+s-1} F when forms have a common (n-1)-fold factor.

Explicit computation of the invariant in specific cases.

Provides necessary conditions for sets of Pfister forms to have a common factor.

Abstract

We study the necessary conditions for sets of quadratic $n$-fold Pfister forms to have a common $(n-1)$-fold Pfister factor. For any set $S$ of $n$-fold Pfister forms generating a subgroup of $I_q^n F/I_q^{n+1} F$ of order $2^s$ in which every element has an $n$-fold Pfister representative, we associate an invariant in $I_q^{n+1} F$ which lives inside $I_q^{n+s-1} F$ when the forms in $S$ have a common $(n-1)$-fold Pfister factor. We study the properties of this invariant and compute it explicitly in a few interesting cases.