Planar vector fields without invariant algebraic curves

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.