Multiplicity of invariant algebraic curves in polynomial vector fields

Multiplicity of Invariant Algebraic Curves and Darboux Integrability

We define four different kinds of multiplicity of an invariant algebraic curve for a given polynomial vector field and investigate their relationships. After taking a closer look at the singularities and at the line of infinity, we improve the Darboux theory of integrability using these new notions of multiplicity.

Algebraic invariant curves of plane polynomial differential systems

We consider a plane polynomial vector field P(x, y) dx +Q(x, y) dy of degree m > 1. With each algebraic invariant curve of such a field we associate a compact Riemann surface with the meromorphic differential ω = dx/P = dy/Q. The asymptotic estimate of the degree of an arbitrary algebraic invariant curve is found. In the smooth case this estimate has already been found by Cerveau and Lins Neto ...

Darboux integrability and invariant algebraic curves for planar polynomial systems

In this paper we study the normal forms of polynomial systems having a set of given generic invariant algebraic curves. PACS numbers: 02.30.Ik, 02.10.De

Algebraic Invariant Curves and Algebraic First Integrals for Riccati Polynomial Differential Systems

We characterize the algebraic invariant curves for the Riccati polynomial differential systems of the form x′ = 1, y′ = a(x)y+ b(x)y+ c(x), where a(x), b(x) and c(x) are arbitrary polynomials. We also characterize their algebraic first integrals.

Vector bundles over algebraic curves