$$Signature(B_{\phi})=\textup{Number of real branches of}\;\; C$$

This result was further generalised to the case of arbitrary Gorenstein curve singularities $C$ in [@jamesduco].

In a similar vein, the work [@sza1] associates to a polynomial mapping $f:{{\mathbb R}}^n {\longrightarrow}{{\mathbb R}}^k$ an ${{\mathbb R}}$-algebra ${\cal A}$ with a quadratic form $B_{\phi}$ such that $$Signature(B_{\phi})=\chi(f^{-1}(0)),$$ and further variations can be found in [@sza2], [@sza3], [@dutertre].\
Another type of application is to the topology of real hypersurface singularities. A function $f \in P$ defines an isolated hypersurface singularity precisely when the partial derivatives $f_i:=\partial f/\partial x_i$ form a regular sequence in $P$. In this case the algebra ${\cal A}$ is nothing but the (real) Milnor algebra $P/J_f$, where $J_f$ is the jacobian ideal of $f$. The degree of $F:=(\partial_1f,\ldots,\partial_nf)$ is the Poincaré-Hopf index of the gradient vector field of $f$.

The Milnor ring is the most important algebraic invariant of the singularity $f$. Its dimension $\mu(f):=\dim(P/J_f)$ is the Milnor number, which equals the dimension of the cohomology $H^{n-1}(F)$ of the Milnor fibre $F$, [@agv], [@milnor]. In the real case one can also consider the real Milnor fibres: $$F_{\epsilon,\eta}:=\{ x \in {{\mathbb R}}^n \;|\; \|x\| \le \epsilon, f(x)=\eta\}, \;\; 0 < \epsilon \ll 1,\;\; 0 < 