Number of invariant straight lines for homogeneous polynomial vector fields of arbitrary degree and dimension
The authors consider in \(\mathbb{R}^d\) (respectively \(\mathbb{C}^d\)) the real (respectively complex) polynomial differential system of degree \(m\): \[ \begin{aligned} \dot x_1 & =P_1(x_1,\dots,x_d), \\ &\quad \cdots\cdots \\ \dot x_d & =P_d(x_1,\dots,x_d),\end{aligned} \] where \(m\geq1\) is the maximum of the degrees of the polynomials \(P_i\) for \(i=1,\dots,d\). In the case that all polynomials \(P_i\) are homogeneous of degree \(m\), they determine the number of invariant straight lines through the origin based on Bézout's theorem in algebraic geometry, when this number is finite. They also show that this number provides an upper bound for the number of infinite singular points of the polynomial differential systems of degree \(m\) in \(\mathbb{R}^d\), and this upper bound is reached if all the invariant straight lines through the origin are real.
scientific article; zbMATH DE number 5615000