Cofactors and equilibria for polynomial vector fields (Q2877946)

The authors deal with complex polynomial vector fields NEWLINE\[NEWLINE \dot x = P(x,y), \;\;\dot y = Q(x,y), \eqno(1) NEWLINE\]NEWLINE where \(P,Q \in \mathbb{C}[x,y]\) are coprime. The main result is a relationship between the existence of equilibrium points of (1) and the cofactors of the invariant algebraic curves and the exponential factors of (1).NEWLINENEWLINEThe main definitions are the following. By \(\mathcal{X}\) denote the first order differential operator associated with (1). The algebraic curve \(f=0\), \(f \in \mathbb{C}[x,y]\), is called \textit{invariant} (w.r.t. (1)) if there exists \(K \in \mathbb{C}[x,y]\), called the \textit{cofactor of} \(f\), such that \(\mathcal{X}f = K f\). The function \(F=e^{g/f}\), where \(f,g \in \mathbb{C}[x,y]\) are coprime, \(f \not\equiv 0\), is called an \textit{exponential factor of} (1) if there exists \(L \in \mathbb{C}[x,y]\), \(\deg L < \max \{\deg P, \deg Q\}\), called the \textit{cofactor of} \(F\), such that \(\mathcal{X}F = L F\). Then \(f=0\) is an invariant algebraic curve of (1) if \(\deg f >0\).