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An Isaac Newton Institute Workshop

Model Theory, Algebraic and Analytic Geometry

Non-oscillating trajectories of vector fields and Zariski's local uniformization.

Authors: cano, felipe (u-valladolid), moussu, robert (u-bourgogne), rolin, jean philippe (u-bourgogne), roche, claude (u-paul sabatier)


A transcendent non-oscillating trajectory of an analytic germ of real vector field induces a structure of Hardy field for the meromorphic functions. It has a natural valuation associated to it. The study of this valuation allows to get a reduction of singularities of the vector field following the strict transform of the trajectory. More generally, for a holomorphic complex vector field, the above results can be generalized for a given valuation of the field of the meromorphic functions. We obtain in this way a local uniformization in the sense of Zariski, that should be globalized in dimension three, following the classical results of Zariski. The key of these results is a construction (due to J. Cano and Grigoriev-Singer) based on the Newton Polygon of a differential operator, that assures finiteness results on the valuation allowing the local uniformization.