Extension of Teissier's results to non-isolated singularities

Let $f:({\mathbb C}^n,0)\rightarrow({\mathbb C},0)$ be a holomorphic function, with $n\ge 2$, and allow $f$ to have non-isolated singularities. The quantities $Q(f)$, $l_0(\operatorname{grad} f)$, and $\rho_0(f)$, as well as the notions of polar branches and topological, analytic, and Lipschitz equivalence, are understood as in Teissier's results above. Extension of Teissier's results. Teissier's results (a--d) hold for $f$ with non-isolated singularities, provided in part (c) that the generic polar locus is non-empty (and hence is a curve). This extends the stated equalities, invariance results, and realization of the Łojasiewicz exponents from isolated to non-isolated singularities for every $n\ge 2$.