The limit cycle of the van der Pol equation is not algebraic (Q1343703)

It is shown that no solution curve of the so-called Liénard equation \((y'' + f(y)y' + g(y) = 0)\) lies in an algebraic curve if \(f\) and \(g\) are nonzero polynomials, if \(\deg f \geq \deg g\) and if \(f/g\) is not constant. The proof is based on a curious but elementary lemma that gives a necessary and sufficient condition for a solution curve of a polynomial system of the first order to be a portion of an irreducible algebraic curve. Then a clever degree calculation leads to the result.