Abstract
The main result of the formal theory of overdetermined systems of differential equations says that any regular system Au = f with smooth coefficients on an open set U ⊂ ℝn admits a solution in smooth sections of the bundle of formal power series provided that f satisfies a compatibility condition in U. Our contribution consists in detailed study of the dependence of formal solutions on the point of the base U of the bundle. We also parameterize these solutions by their Cauchy data. In doing so, we prove that, under absence of topological obstructions, there is a formal solution which smoothly depends on the point of the base. This leads to a concept of a finitely generated system (do not mix up it with holonomic or finite -type systems) for which we then prove a C ∞-Poincaré lemma.
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Shlapunov, A.A., Tarkhanov, N.N. A homotopy operator for Spencer’s sequence in the C ∞-case. Sib. Adv. Math. 19, 91–127 (2009). https://doi.org/10.3103/S1055134409020035
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DOI: https://doi.org/10.3103/S1055134409020035