Hyperelliptic continued fractions

25 October 2021
Francesco Ballini

We can define a continued fraction for formal series $f(t)=\sum_{i=-\infty}^d c_it^i$ by repeatedly removing the polynomial part, $\sum_{i=0}^d c_it^i$, (the equivalent of the integer part) and inverting the remaining part, as in the real case. This way, the partial quotients are polynomials. Both the usual continued fractions and the polynomial continued fractions carry properties of best approximation. However, while for square roots of rationals the real continued fraction is eventually periodic, such periodicity does not always occur for $\sqrt{D(t)}$. The correct analogy was found by Abel in 1826: the continued fraction of $\sqrt{D(t)}$ is eventually periodic if and only if there exist nontrivial polynomials $x(t)$, $y(t)$ such that $x(t)^2-D(t)y(t)^2=1$ (the polynomial Pell equation). Notice that the same holds also for square root of integers in the real case. In 2014 Zannier found that some periodicity survives for all the $\sqrt{D(t)}$: the degrees of their partial quotients are eventually periodic. His proof is strongly geometric and it is based on the study of the Jacobian of the curve $u^2=D(t)$. We give a brief survey of the theory of polynomial continued fractions, Jacobians and an account of the proof of the result of Zannier.

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