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.

翻译：我们证明，对每个实奇点 $f: (\mathbb{R}^{n+1}, 0) \to (\mathbb{R}, 0)$，可关联两个微分方程组 $\mathfrak{g}^{k\pm}_f$，它们分别是正、负Milnor纤维化全空间上实解析函数层在 $\mathcal{D}$-模范畴中沿 $\mathbb{R}^{\pm}$ 的前推。我们证明，当 $k=0$ 且 $f$ 为孤立奇点时，$\mathfrak{g}^{\pm}$ 决定了正、负Milnor纤维的第 $n$ 同调群。随后，我们计算了普通二次奇点的 $\mathfrak{g}^{+}$，并证明在特定扰动化选择条件下，可恢复任意孤立奇点 $f$ 的Milnor纤维的拓扑最大同调群。作为应用，我们基于奇点扰动化构造了一种公钥加密方案。