Inflection curves of rational vector fields

Abstract

We initiate the study of inflection curves of rational vector fields on the Riemann sphere. For a rational vector field $v_R=-R(z)\frac{\partial}{\partial z}, \qquad R(z)=\frac{Q(z)}{P(z)} $ we define its affine regular inflection locus by $ \{z\in \mathbb{C}: R(z)\ne0,\ P(z)\neq 0,\ \text{Im} R'(z)=0\} $ and its algebraic inflection curve by the closure of this locus, equivalently by $ \mathfrak{I}_R=(R')^{-1}(\mathbb{RP}^1). $ We prove an explicit defining equation, degree bounds, local normal forms near poles, the asymptotic directions at infinity, and a maximum-principle obstruction excluding compact components without poles. We also explain that these curves are precisely the real dessins associated with exact rational differentials, i.e. rational differentials with zero residues. Finally, we give a reducibility criterion for the complexification, prove a generic irreducibility statement in the usual separated-variable sense, and classify the exact dessins of degree at most two. The degree-three problem is reduced to three explicit normal forms.