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Vector fields, invariant varieties and linear systems. (English) Zbl 1107.37038

Summary: We investigate the interplay between invariant varieties of vector fields and the inflection locus of linear systems with respect to the vector field. Among the consequences of such investigation we obtain a computational criterion for the existence of rational first integrals of a given degree, bounds for the number of first integrals on families of vector fields, and a generalization of Darboux’s criteria. We also provide a new proof of Gomez-Mont’s result on foliations with all leaves algebraic.

MSC:

37F75 Dynamical aspects of holomorphic foliations and vector fields
32M25 Complex vector fields, holomorphic foliations, \(\mathbb{C}\)-actions
34M45 Ordinary differential equations on complex manifolds
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