On an Extension of Hoffmann’s Separation Theorem for Quadratic Forms

Abstract

AbstractLet $p$ and $q$ be anisotropic non-degenerate quadratic forms of dimension $\geq 2$ over an arbitrary field $F$, let $s$ be the unique non-negative integer for which $2^s<{\textrm{dim}( p)} \leq 2^{s+1}$, and let $k$ be the dimension of the anisotropic part of $q$ after extension to $F(p)$. A recent conjecture of the author then asserts that ${\textrm{dim}( q)}$ must lie within $k$ of an integer multiple of $2^{s+1}$. This statement, which holds trivially if $k \geq 2^s -1$, represents a natural generalization of the well-known separation theorem of Hoffmann, bridging a gap between it and certain classical results on the Witt kernels of function fields of quadrics. In the present article, we prove the conjecture in the case where $\textrm{char}(F) \neq 2$ and ${\textrm{dim}( p)}> 2k - 2^{s-1}$. This implies, in particular, that the conjecture holds if $\textrm{char}(F) \neq 2$ and either $k \leq 2^{s-1} + 2^{s-2}$ or ${\textrm{dim}( p)} \geq 2^s + 2^{s-1} - 4$.