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Polynomial curves on trinomial hypersurfaces

Volume 186 / 2018

Ivan Arzhantsev Acta Arithmetica 186 (2018), 87-99 MSC: Primary 14M20, 14R20; Secondary 11D41, 14J50. DOI: 10.4064/aa170813-18-2 Published online: 19 September 2018


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.


  • Ivan ArzhantsevNational Research University Higher School of Economics
    Faculty of Computer Science
    Kochnovskiy Proezd 3
    Moscow, 125319 Russia

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