Darboux theory of integrability for real polynomial vector fields on the n-dimensional ellipsoid

2024/10/27 by Llibre, J., Murza, Adrian C.
#34C05 #34C07 #34C40 #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2410.21336

Abstract

We extend to the n-dimensional ellipsoid contained in \Rⁿ⁺¹, the Darboux theory of integrability for polynomial vector fields in the n-dimensional sphere (Llibre et al., 2018). New results on the maximum number of invariant parallels and meridians of polynomial vector fields \X on the invariant n-dimensional ellipsoid, as a function of its degree, are provided. Our results extend the known result on the upper bound for the number of invariant hyperplanes that a polynomial vector field \Y in \Rⁿ can have in function of the degree of \Y.

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