On the Gauss-Manin Connection and Real Singularities

Abstract

We prove that to each real singularity $f:({\mathbb{R}}^{n+1},0)\to (\mathbb{R},0)$ one can associate two systems of differential equations ${\mathfrak{g}}_f^{k\pm }$ which are pushforwards in the category of $\mathcal{D}$-modules over ${\mathbb{R}}^{\pm }$, of the sheaf of real analytic functions on the total space of the positive, respectively negative, Milnor fibration. We prove that for $k=0$ if $f$ is an isolated singularity then ${\mathfrak{g}}^{\pm }$ determines the the $n$-th homology groups of the positive, respectively negative, Milnor fibre. We then calculate ${\mathfrak{g}}^{+}$ for ordinary quadratic singularities and prove that under certain conditions on the choice of morsification, one recovers the top homology groups of the Milnor fibers of any isolated singularity $f$. As an application we construct a public-key encryption scheme based on morsification of singularities.