Zariski equisingularity of surface singularities in ${\mathbb{C}}^3$ by a local invariant

We associate to every analytic surface singularity $(V,0)$ in $({\mathbb{C}}^3,0)$, not necessarily isolated, an invariant $mul{t}^{\ast }(V)$ and show that an analytic family of such singularities $(V_t,0)$, $t\in ({\mathbb{C}}^l,0)$, is generically Zariski equisingular if and only if $mul{t}^{\ast }(V_t)$ is constant. The invariant, that we call the multiplicity sequence of $V$, takes into account the multiplicities of the successive discriminants of $V$ by generic corank one projections.