Linkage of Pfister forms over semi-global fields

We study linkage of ( d + 1 ) -fold quadratic Pfister forms over function fields in one variable over a henselian valued field of 2-cohomological dimension d . Specifically, we characterise this property in terms of linkage of quadratic Pfister forms over function fields over the residue field of the henselian valued field; in full generality in characteristic different from 2, and for most complete discretely valued fields in characteristic 2. As an application, we obtain a proof that ( d + 2 ) -fold quadratic Pfister forms over function fields in one variable over a d -dimensional higher local field are linked.