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On an extension of the separation theorem for quadratic forms over fields Scully, Stephen


The problem of determining conditions under which a rational map can exist between a pair of twisted flag varieties plays an important role in the study of algebraic groups and their torsors over general fields. A non-trivial special case arising in the theory of quadratic forms amounts to studying the extent to which an anisotropic quadratic form can become isotropic after extension to the function field of a quadric. To this end, let $p$ and $q$ be anisotropic quadratic forms over an arbitrary field, and let $k$ be the dimension of the anisotropic part of $q$ over the function field of the quadric $p=0$. We then conjecture that the dimension of $q$ lies within $k$ of an integer multiple of $2^{s+1}$, where $2^{s+1}$ is the least power of 2 bounding the dimension of $p$ from above. This generalizes the so-called ``separation theorem'' of D. Hoffmann, bridging the gap between it and some other classical results on Witt kernels of function fields of quadrics. The statement holds trivially if $k \geq 2^s - 1$. In this talk, I will discuss recent work that confirms its validity in the case where $k \leq 2^{s-1} + 2^{s-2}$ (among other cases).

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