Polynomial curves on trinomial hypersurfaces
We prove that every rational trinomial affine hypersurface admits a horizontal polynomial curve. This result provides an explicit non-trivial polynomial solution to a trinomial equation. Also we show that a trinomial affine hypersurface admits a Schwarz–Halphen curve if and only if the trinomial comes from a platonic triple. It is a generalization of Schwarz–Halphen’s Theorem for Pham–Brieskorn surfaces.