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On the polynomial vector fields on \(\mathbb S^3\). (English) Zbl 1241.34038

Let \(X\) be a polynomial vector field of degree \(n\) on \(M\), \(M=\mathbb{R}^{m}\). The dynamics and the algebraic-geometric properties of the vector fields \(X\) have been studied, mainly, for the case \(M=\mathbb{R}^{2}\) and for homogeneous polynomial vector fields of degree \(n\) on \(\mathbb{S}^{2}\). There are very few results on the non-homogeneous polynomial vector fields of degree \(n\) on \(\mathbb{S}^{3}\). The paper attempts to rectify this slightly.

MSC:

34C08 Ordinary differential equations and connections with real algebraic geometry (fewnomials, desingularization, zeros of abelian integrals, etc.)
34C07 Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
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