On Meromorphic Parameterizations of Real Algebraic Curves

A singular flat geometry may be canonically assigned to a real algebraic curve Γ; namely, via analytic continuation of unit speed parameterization of the real locus ΓR. Globally, the metric ρ = |Q| = |q(z)|dzdz̄ is given by the meromorphic quadratic differential Q on Γ induced by the standard complex form dx + dy on C = {(x, y)}. By considering basic properties of Q, we show that the condition for local arc length parameterization along ΓR to extend meromorphically to the complex plane is quite restrictive: For curves of degree at most four, only lines, circles and Bernoulli lemniscates have such meromorphic parameterizations.