On the Gauss-Manin Connection and Real Singularities

TL;DR

This paper establishes a connection between real singularities and differential equations, showing how to recover topological invariants of Milnor fibers and applying this to a new cryptographic scheme.

Contribution

It introduces systems of differential equations associated with real singularities, linking them to Milnor fiber homology and proposing a novel encryption method based on singularity morsification.

Findings

Systems of differential equations encode Milnor fiber homology.

Explicit calculation for quadratic singularities.

Encryption scheme based on singularity morsification.

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 $\mathfrak{g}^{k\pm}_f$ 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.