Binary quadratic refinement ($Q_v(x)=Q_0(x)+B(v,x)$)

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 $ϵ(Q_v)=ϵ(Q_0)(-1{)}^{Q_0(v)}$.