On the Gauss-Manin Connection and Real Singularities
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866909083183874048 |
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| author | Andersen, Lars |
| author_facet | Andersen, Lars |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_06849 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On the Gauss-Manin Connection and Real Singularities Andersen, Lars Algebraic Geometry Information Theory 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. |
| title | On the Gauss-Manin Connection and Real Singularities |
| topic | Algebraic Geometry Information Theory |
| url | https://arxiv.org/abs/2206.06849 |