= Binary quadratic refinement
{title2=$Q_v(x)=Q_0(x)+B(v,x)$}
A <binary quadratic refinement> of a fixed nonsingular <alternating bilinear form> $B$ has polar form $B$. The difference of two refinements is linear, so nondegeneracy gives the displayed unique parametrization by $v$. For $\dim V=2m$, refinements of plus and minus <Arf invariant of a quadratic form> have respectively $2^{m-1}(2^m+1)$ and $2^{m-1}(2^m-1)$ elements. Completing the square in the binary character sum gives $\epsilon(Q_v)=\epsilon(Q_0)(-1)^{Q_0(v)}$.
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