Non-algebraicity of foliations via reduction modulo $2$

Motivated by the Jouanolou foliation problem, we investigate the non-algebraicity of foliations by curves on ${\mathbb{P}}_{\mathbb{C}}^2$. We present a criterion to show that such a foliation has no algebraic invariant curves, using a method of reduction modulo $2$. Finally, using this criterion, we give a new proof that the Jouanolou foliation of odd degree has no algebraic invariant curves. We also present other classes of foliations without algebraic invariant curves.