Linkage of Pfister forms over $\mathbb C(x_1,\ldots,x_n)$

Abstract

We prove the existence of a set of cardinality $2^n$ of $n$-fold Pfister forms over $ℂ\left(x_1,\dots ,x_n\right)$ which do not share a common $\left(n-1\right)$-fold factor. This gives a negative answer to a question raised by Becher. The main tools are the existence of the dyadic valuation on the complex numbers and recent results on symmetric bilinear forms over fields of characteristic 2.