Polynomial curves on trinomial hypersurfaces
Volume 186 / 2018
Acta Arithmetica 186 (2018), 87-99
MSC: Primary 14M20, 14R20; Secondary 11D41, 14J50.
DOI: 10.4064/aa170813-18-2
Published online: 19 September 2018
Abstract
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.
