Twin quadratic polynomial vector fields in the space \(\mathbb{C}^3\)
{\em Twin vector fields} are two different vector fields that agree on position and spectra at all their singularities. In several previous papers this notion has been studied for planar quadratic vector fields and for some vector fields in dimension \(3\). In the present paper the authors study this notion for quadratic vector fields in \(\mathbb{C}^3\) having exactly \(8\) singular points. By Bézout's Theorem, eight is the maximum number of isolated singularities that a quadratic vector field in \(\mathbb{C}^3\) can have and, thus, all the singularities are non-degenerated, that is, their associated eigenvalues are all different from zero. \N\NIn this family of vector fields, the authors fix as a hypothesis the position and spectra of \(M\) singular points and also some conditions on their associated eigenvalues. They study the cases when \(M=7\), \(M=6\) and \(M=5\). They also provide a theorem establishing the existence of twin vector fields inside the considered family. One of the main tools in the proof of their results is Euler-Jacobi formula.
scientific article; zbMATH DE number 7875573