Local-global principle for quadratic forms over fraction fields of two-dimensional henselian domains

Let $R$ be a 2-dimensional normal excellent henselian local domain in which 2 is invertible and let $L$ and $k$ be respectively its fraction field and residue field. Let $Ω_R$ be the set of rank 1 discrete valuations of $L$ corresponding to codimension 1 points of regular proper models of $\Spec R$. We prove that a quadratic form $q$ over $L$ satisfies the local-global principle with respect to $Ω_R$ in the following two cases: (1) $q$ has rank 3 or 4; (2) $q$ has rank $\ge 5$ and $R=A[y]$, where $A$ is a complete discrete valuation ring with a not too restrictive condition on the residue field $k$, which is satisfied when $k$ is $C_1$.

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