The Hilbert-Arnold problem and an estimate of the cyclicity of polycycles of the plane and in space (Q5951274)

This paper is motivated by the Hilbert-Arnold problem about the uniform boundedness of the number of limit cycles for generic families of vector fields on the sphere \(S^2\). The elementary cyclicity \(E(n)\) is the maximum possible cyclicity of a polycycle occuring in a generic \(n\)-parameter family of \(C^\infty\) vector fields \(S^2\). The author proves that \(E(n)\leq 2^{25n^2}\) holds for each positive integer \(n\). Moreover, the author also presents an estimate of the cyclicity of a multidimensional polycycle.