  We introduce bilinear forms in a flag in a complete intersection local
$\mathbb R$-algebra of dimension 0, related to the Eisenbud-Levine,
Khimshiashvili bilinear form. We give a variational interpretation of these
forms in terms of Jantzen's filtration and bilinear forms. We use the
signatures of these forms to compute in the real case the constant relating the
GSV-index with the signature function of vector fields tangent to an even
dimensional hypersurface singularity, one being topologically defined and the
other computable by finite dimensional commutative algebra methods.
