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On the Gauss-Manin Connection and Real Singularities (2206.06849v3)

Published 14 Jun 2022 in math.AG, cs.IT, and math.IT

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.

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