Local Uniformization of Codimension One Foliations
Miguel Fern\'andez-Duque
TL;DR
This paper proves the existence of local uniformization for codimension one foliations in any dimension, specifically for rational archimedean valuations, extending known results from lower dimensions.
Contribution
It establishes the first general proof of local uniformization for codimension one foliations in arbitrary dimensions for rational archimedean valuations.
Findings
Proves local uniformization exists in all dimensions for these foliations.
Extends classical results from 2 and 3 dimensions to higher dimensions.
Addresses an open problem in the theory of foliations.
Abstract
The reduction of singularities of codimension one foliations is known in the case of ambient dimension 2 (Seidenberg, A. (1968). Reduction of singularities of the differential equation Ady= Bdx. American Journal of Mathematics, 90(1), 248-269) and 3 (Cano, F. (2004). Reduction of the singularities of codimension one singular foliations in dimension three. Annals of mathematics, 160(3), 907-1011). However, in greater ambient dimension it is still an open problem. In this work we prove the existence of local uniformization for codimension one foliations in arbitrary ambient dimension, in the case of rational archimedean valuations.
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Advanced Differential Equations and Dynamical Systems · Polynomial and algebraic computation