论文标题
在高斯 - 曼宁的连接和真正的奇点上
On the Gauss-Manin Connection and Real Singularities
论文作者
论文摘要
我们证明了每个真正的奇异性$ f：（\ mathbb {r}^{n+1}，0）\与（\ mathbb {r}，0）$一个人可以关联两个微分方程的系统$ \ mathfrak {g}^{g}^{k \ pm}^{k \ pm} _f $ $ nath $ nath $ - $ \ mathbb {r}^{\ pm} $，在正面的负面，负，milnor纤维的总空间上的真实分析函数的捆绑。我们证明，对于$ k = 0 $，如果$ f $是一个孤立的奇异性，则$ \ mathfrak {g}^{\ pm} $确定$ n $ th $ th $ th $ th $ n $ th的同源组，分别为负，milnor纤维。然后，我们为普通二次奇异点计算$ \ mathfrak {g}^{+} $，并证明在某些条件下，人们在选择杂交的情况下恢复了任何孤立的奇异性$ f $的Milnor纤维的顶级同源组。作为应用程序，我们基于奇异性的层化构建了公钥加密方案。
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.
