Pell surfaces

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In 1826 Abel started the study of the polynomial Pell equationx2 − g(u)y2 = 1. Its solvability in polynomials x(u), y(u) depends on a certain torsionpoint on the Jacobian of the hyperelliptic curve v2 = g(u). In this paper westudy the affine surfaces defined by the Pell equations in 3-space with coordinatesx, y, u, and aim to describe all affine lines on it. These are polynomial solutions ofthe equation x(t)2 − g(u(t))y(t)2 = 1. Our results are rather complete when thedegree of g is even but the odd degree cases are left completely open. For evendegrees we also describe all curves on these Pell surfaces that have only 1 placeat infinity.

Original languageEnglish (US)
Pages (from-to)478-518
Number of pages41
JournalActa Mathematica Hungarica
Issue number2
StatePublished - Apr 1 2020

All Science Journal Classification (ASJC) codes

  • General Mathematics


  • Pell equation
  • affine line
  • algebraic surface


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