Transcendental limit cycles via the structure of arbitrary degree invariant algebraic curves of polynomial planar vector fields

In this paper we consider planar polynomial vector fields and we show certain structure that their invariant algebraic curves should have. This approach allows to obtain results on the nonexistence of those algebraic curves for arbitrary degree. As an application of this algorithmic method we easily prove that the van der Pol oscillator cannot have any algebraic solution and in particular neither is his limit cycle algebraic. In addition we show that a limit cycle studied by Dolov is not algebraic.