Planar vector fields without invariant algebraic curves

TL;DR

This paper extends methods for detecting invariant algebraic curves in polynomial vector fields, showing that generically the coordinate axes are the only such curves and refining bounds related to Hilbert's 16th problem.

Contribution

It introduces a residual set framework for the uniqueness of invariant algebraic curves and applies it to Kolmogorov vector fields, refining classical results.

Findings

Coordinate axes are generically the only invariant algebraic curves in Kolmogorov vector fields.

The space of vector fields with a prescribed invariant curve is residual and of full measure.

Existence of bounds for limit cycles can be achieved without algebraic limit cycles.

Abstract

In this work we revisit and extend the method introduced by Lins Neto, Sad and Sc\'{a}rdua for detecting the non-existence of invariant algebraic curves other than some prescribed invariant nodal curve. We prove that, under the existence of a suitable example, the space of polynomial vector fields whose elements have the prescribed curve as their unique invariant algebraic curve is residual and of full measure. We apply this framework to Kolmogorov vector fields, showing that generically the coordinate axes are the unique invariant algebraic curves. Finally, we also refine existing characterizations related to Hilbert's 16th problem, showing that if there exists a bound for the number of limit cycles of a vector field of degree n, then it can be attained by a vector field without algebraic limit cycles.