On the algebraic invariant curves of plane polynomial differential systems

Details On the algebraic invariant curves of plane polynomial differential systems On the algebraic invariant curves of plane polynomial differential systems

2000-09-22

arxiv.org/abs/math/0009208v3

Mathematics

Dynamical Systems

10 pages, Latex, to appear in J.Phys.A:Math.Gen

Scientific paper

10.1088/0305-4470/34/3/325

We consider a plane polynomial vector field $P(x,y)dx+Q(x,y)dy$ of degree $m>1$. To each algebraic invariant curve of such a field we associate a compact Riemann surface with the meromorphic differential $\omega=dx/P=dy/Q$. The asymptotic estimate of the degree of an arbitrary algebraic invariant curve is found. In the smooth case this estimate was already found by D. Cerveau and A. Lins Neto [Ann. Inst. Fourier Grenoble 41, 883-903] in a different way.

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