Index of Singularities of Real Vector Fields on Singular Hypersurfaces

G'omez-Mont, Seade and Verjovsky introduced an index, now called GSV-index, generalizing the Poincar'e-Hopf index to complex vector fields tangent to singular hypersurfaces. The GSV-index extends to the real case. This is a survey paper on the joint research with G'omez-Mont and Giraldo about calculating the GSV-index ${\text{Ind}}_{V_{\pm },0}(X)$ of a real vector field $X$ tangent to a singular hypersurface $V=f^{-1}(0)$. The index ${\text{Ind}}_{V_{\pm ,0}}(X)$ is calculated as a combination of several terms. Each term is given as a signature of some bilinear form on a local algebra associated to $f$ and $X$. Main ingredients in the proof are G'omez-Mont’s formula for calculating the GSV-index on singular complex hypersurfaces and the formula of Eisenbud, Levine and Khimshiashvili for calculating the Poincar'e-Hopf index of a singularity of a real vector field in ${\mathbb{R}}^{n+1}$