For a field F of characteristic different from 2, containing a square root of -1, endowed with an F~(×2)-compatible valuation v such that the residue field has at most two square classes, we use a combinatorial analogue of the Witt ring of F to prove that an anisotropic quadratic form over F with even dimension d, trivial discriminant, and Hasse-Witt invariant can be written in the Witt ring as the sum of at most (d~2)/8 3-fold Pfister forms.
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