Abstract

In this work invariant algebraic curves of a planar polynomial differential system play the fundamental role. If an irreducible invariant algebraic curve for a planar polynomial differential system exists, then the values of its cofactor at each non-degenerate singular point are determined. In fact, we will show that this value is a linear combination with natural coefficients of the eigenvalues associated to the non-degenerate singular point. These natural coefficients can be completely determined in some cases depending on the nature of the singular point. Moreover, the points at infinity can also be taken into account. Once the system is considered in the projective complex plane, the degree of an invariant algebraic curve becomes a parameter of its cofactor. If we consider a system of degree $d$, then it has $d2+d+1$ singular points (counted with multiplicity) and the cofactor of an invariant algebraic curve is a polynomial of degree at most $d-1$. We proceed as follows: we take a polynomial of degree $d-1$ with its $d(d+1)/2$ arbitrary coefficients and we assume that it is the cofactor of an irreducible invariant algebraic curve of degree $n$. Then, we impose all the conditions given by the non-degenerate singular points. In most cases, we impose $d2+d+1$ conditions and, hence, we completely determine the cofactor and the degree of the curve, whose existence can be determined by solving a linear system of equations, or we show an incompatibility condition. Therefore, we can determine the existence of all the invariant algebraic curves in many cases. We will also show several applications of these conditions. First, we state that all the known quadratic systems with an algebraic limit cycle only exhibit one invariant algebraic curve, whose oval is the limit cycle. We will present a particular example of the proof of this result. To finish with, we show several formulae relating the degree of a system whose singular points are all simple, the eigenvalues related to all its singular points and the degree of an invariant algebraic curve. These formulae have been proved in a work of Jean Moulin-Ollagnier mainly using the explained ideas.