Let $f:(\mathbb{C}^n,0)\to (\mathbb{C},0)$ be a reduced complex analytic function. We write $$f=f_m+f_{m+1}+\cdots+f_k+\cdots$$ where each $f_k$ is a homogeneous polynomial of degree $k$ and $f_m\neq 0$. We have the decomposition of $f_m$ in irreducible polynomials $$f_m=h_1^{k_1}\cdots h_r^{k_r}.$$ Thus, we define the multiplicity of $V(f)$ along of $V(h_i)$ for $k_{V(f)}(V(h_i)):=k_i$. The numbers $k_{V(f)}(V(h_1)),..., k_{V(f)}(V(h_r))$ are called, as well, the multiplicities relatives of $V(f)$.
Let $f,g:(\mathbb{C}^n,0)\to (\mathbb{C},0)$ be two reduced complex analytic functions. It is well know that this numbers are topological invariants when $n=2$, i.e., if there is a homeomorphism $\varphi:(\mathbb{C}^2,V(f),0)\to (\mathbb{C}^2,V(f),0)$, then reordering the indices, if necessary, $k_{V(f)}(X_i)=k_{V(g)}(Y_i)$, for $i=1,...,r$, where $X_1,...,X_r$ (resp. $Y_1,...,Y_r$) are the irreducible components of the tangent cone of $V(f)$ (resp. $V(g)$) at the origin.
Is this result true for $n>2$?
