On the Gauss-Manin Connection and Real Singularities
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We prove that to each real singularity f: (ℝ^n+1, 0) → (ℝ, 0) one can associate two systems of differential equations 𝔤^k±_f which are pushforwards in the category of 𝒟-modules over ℝ^±, 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 𝔤^± determines the the n-th homology groups of the positive, respectively negative, Milnor fibre. We then calculate 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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