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Given a real analytic vector field tangent to a hypersurface with an algebraically isolated singularity we introduce a relative Jacobian determinant in the finite dimensional algebra associated with the singularity of the vector field on . We show that the relative Jacobian generates a 1-dimensional non-zero minimal ideal. With its help we introduce a non-degenerate bilinear pairing, and its signature measures the size of this point with sign. The signature satisfies a law of conservation of...
We study pseudo-abelian integrals associated with polynomial deformations of slow-fast Darboux integrable systems. Under some assumptions we prove local boundedness of the number of their zeros.
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