of Eisenbud-Levine [@eisenbudlevine], and Khimshiashvilli [@khimshiashvilli] provides an algebraic method to determine the local degree of a finite map germ $F:({{\mathbb R}}^n,0){\longrightarrow}({{\mathbb R}}^n,0)$ with component functions $f_1,f_2,\ldots,f_n \in P:={{\mathbb R}}[[x_1,x_2,\ldots,x_n]]$. One considers the local ${{\mathbb R}}$-algebra $${\cal A}_F:=P/(f_1,f_2,\ldots,f_n),$$ which has finite dimension precisely when $f_1,f_2,\ldots,f_n$ form a regular sequence in $P$. Morever, in that case ${\cal A}_F$ is a [*Gorenstein ring*]{}: if we let $$h:=\left| \begin{array}{ccc} \partial_1f_1&\ldots&\partial_n f_1\\
\vdots& &\vdots\\\partial_1 f_n& \ldots&\partial_n f_n \\
\end{array} \right|$$ and choose any linear form $\phi: {\cal M} {\longrightarrow}{{\mathbb R}}$ with $\phi(h) \neq 0$, then the pairing $$B_{\phi}: {\cal A}_F \times {\cal A}_F {\longrightarrow}{{\mathbb R}},\;(a,b) \mapsto \phi(a\cdot b)$$ is non-degenerate.

(Eisenbud-Levine [@eisenbudlevine], Khimshiashvilli [@khimshiashvilli]) If $\phi(h) >0$ then $$Signature(B_{\phi})=Degree(F,0)$$

The theorem is a result of key importance and has been the starting point of many subsequent works. We mention a few of the applications and generalisations.

Consider an isolated complete intersection curve $C=f^{-1}(0)$, where $f:({{\mathbb R}}^n,0) {\longrightarrow}({{\mathbb R}}^{n-1},0)$. According to Aoki, Fukuda and Nishimura [@aokifukudanishimura], one can compute the [*number of real branches of $C$*]{} as follows: consider $g:=x_1^2+x_2^2+\ldots+x_n^2$ and let $$Jac(g):=\left| \begin{array}{ccc} \partial_1 g&\ldots&\partial_n g\\
\partial_1 f_1&\ldots&\partial_nf_1\\\vdots&&\vdots\\\partial_1f_{n-1}&\ldots&\partial_nf_{n-1}\\ \end{array} \right|$$ On the ${{\mathbb R}}$-algebra ${\cal A}_{f,Jac(g)}:=P/(f_1,\ldots,f_{n-1},Jac(g))$ one defines as above a pairing $B_{\phi}$.

(Aoki-Fukuda-Nishimura) 