The improved Euler-Jacobi formula and the planar polynomial vector fields of degree 2-4

Abstract

The new Euler-Jacobi formula for double points provides an algebraic relation between the singular points of a polynomial vector field and their topological indices. Using this formula we obtain the configuration of the singular points together with their topological indices for the planar quadratic--quartic polynomial differential systems when these systems have seven finite singular points being one of them with multiplicity two. The case with eight finite singular points has already been solved using the classical Euler-Jacobi formula.