Hyperelliptic continued fractions in the singular case of genus zero

  • Francesco Ballini

    University of Oxford, UK
  • Francesco Veneziano

    University of Genoa, Italy
Hyperelliptic continued fractions in the singular case of genus zero cover

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Abstract

Given a polynomial of even degree with complex coefficients, we consider the continued fraction expansion of . In this setting, it has been shown by Zannier that the sequence of the degrees of the partial quotients of the continued fraction expansion of is eventually periodic, even when the expansion itself is not. In this article, we work out in detail the case in which the curve has genus , establishing explicit geometric conditions corresponding to the appearance of partial quotients of certain degrees in the continued fraction expansion. We also show that there are non-trivial polynomials with non-periodic expansions such that infinitely many partial quotients have degree greater than one.

Cite this article

Francesco Ballini, Francesco Veneziano, Hyperelliptic continued fractions in the singular case of genus zero. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 33 (2022), no. 4, pp. 795–832

DOI 10.4171/RLM/989