# Polynomial curves on trinomial hypersurfaces

## TL;DR

This paper proves the existence of horizontal polynomial curves on rational trinomial hypersurfaces and characterizes when Schwartz-Halphen curves exist, generalizing classical results to higher dimensions.

## Contribution

It establishes the existence of explicit polynomial solutions on rational trinomial hypersurfaces and characterizes the conditions for Schwartz-Halphen curves, extending known theorems.

## Key findings

- Every rational trinomial affine hypersurface admits a horizontal polynomial curve.
- A trinomial affine hypersurface admits a Schwartz-Halphen curve if and only if it derives from a platonic triple.
- Generalizes Schwartz-Halphen's Theorem to higher-dimensional trinomial hypersurfaces.

## 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 Schwartz-Halphen curve if and only if the trinomial comes from a platonic triple. It is a generalization of Schwartz-Halphen's Theorem for Pham-Brieskorn surfaces.