of $f$; it equals the signature of the canonical pairing $B:M_f \times M_f {\longrightarrow}{{\mathbb R}}$, where $M_f:=H_{\bf m}^0(\Omega^n/df\wedge \Omega^{n-1})$. Note that, in the definition with $B_f$, if we interchange two coordinates, then $dx_1\wedge\ldots \wedge dx_n$ changes sign, but also the order the $f_i$ is changed in a corresponding way. As a result, the signature of $B_f$ does not change.

It is easy to verify that $\sigma$ has the following properties:\
1) $\sigma(-f)=(-1)^n \sigma(f)$.\
2) If $f$ has at most one-dimensional critical locus, and $g$ has an isolated critical point, then $$\sigma(f \oplus g)=\sigma(f) \cdot \sigma(g)$$ Here $\oplus$ denotes the Thom-Sebastiani sum of $f$ and $g$. Furthermore, one computes $\sigma(x^2)=1$, hence $\sigma(-x^2)=-1$, and so $\sigma(x^2+y^2)=\sigma(-(x^2+y^2)=1$, whereas $\sigma(x^2-y^2)=-1$. For $x^2y \in {{\mathbb R}}[[x,y]]$ one finds $$\sigma(x^2y)=1$$ so that $\sigma(x^2y+z^2)=1$, $\sigma(x^2y-z^2)=-1$. Note that $x^2y-z^2=-(x^2(-y)+z^2)$, but indeed $\sigma(-f)=-\sigma(f)$ for a function of three variables.

\[signatureconstant\] Let $f \in {{\mathbb R}}\{x_1,x_2,\ldots,x_n\}$ be a function with a one-dimensional critical locus $\Sigma$ and let $F:X \times S {\longrightarrow}{{\mathbb R}}\times S$ be a good representative of an admissible deformation of $f,\sigma$. Then for a $s \in S$ one has $$\sigma(f)=\sum_{p \in X} \sigma(F_s,p)$$

We denote by $\pi: X \times S {\longrightarrow}S$ the canonical projection and consider as before 