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Regular version of the site

Article

Polynomial curves on trinomial hypersurfaces

Acta Arithmetica. 2018. Vol. 186. No. 1. P. 87-99.

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.