\eta \ll \epsilon$$ For $\eta >0$, we put $F_{+}:=F_{\epsilon,\eta}$ and $F_{-}:=F_{\epsilon,-\eta}$ and call them the [*positive*]{} and [*negative*]{} Milnor fibers of $f$.

([@arnold]) Let $f \in {{\mathbb R}}\{x_1,x_2,\ldots,x_n\}$ have an isolated critical point at the origin. Then: $$Signature(f)=-\widetilde{\chi}(F_{-})=(-1)^{n-1}\widetilde{\chi}(F_{+}),$$ where $\widetilde{\chi}$ denotes the reduced Euler characteristic.

Although the positive and negative Milnor fibre have, in general, a quite different topology, it is a simple but remarkable fact that their reduced Euler characteristics are the same, up to a sign. The real $D_4$ surface singularity defined by $x(x^2-y^2)+z^2=0$ may exemplify this.

  ----------------------- ------------------- -----------------------
                                              
   negative Milnor fibre   $D_4$ singularity   positive Milnor fibre
  ----------------------- ------------------- -----------------------

In this paper we give a partial generalisation of this result to the case where the sequence $f_1,f_2,\ldots,f_n$ defines a [*one-dimensional locus*]{}. In this case one says the the ideal $J=(f_1,\ldots,f_n)$ defines an [*almost complete intersection*]{}. We show that the torsion sub-module $M=I/J$ of $P/J$ is a [*Gorenstein module*]{}. An isomorphism $\phi: P \stackrel{\cong}{{\longrightarrow}} \Omega^n_P$ determines a natural pairing $$B_{\phi}: I/J \times I/J {\longrightarrow}{{\mathbb R}}$$ Its signature is an invariant of the real topology of the situation.

In the case of partial derivatives $f_i=\partial f/ \partial x_i$ of a function $f$ with a one-dimensional critical locus, such 