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Let $p$ and $q$ be anisotropic quadratic forms over a field $F$ ofcharacteristic $\neq 2$, let $s$ be the unique non-negative integer such that$2^s < \mathrm{dim}(p) \leq 2^{s+1}$, and let $k$ denote the dimension of theanisotropic part of $q$ after scalar extension to the function field $F(p)$ of$p$. We conjecture that $\mathrm{dim}(q)$ must lie within $k$ of a multiple of$2^{s+1}$. This can be viewed as a direct generalization of Hoffmann'sseparation theorem. Among other cases, we prove that the conjecture is true if$k<2^{s-1}$. When $k=0$, this shows that any anisotropic form representing anelement of the kernel of the natural restriction homomorphism $W(F)\rightarrowW(F(p))$ has dimension divisible by $2^{s+1}$.

Hyperbolicity and near hyperbolicity of quadratic forms over function fields of quadrics