Hyperelliptic continued fractions in the singular case of genus zero (Q6111614)

The paper contain useful ideas. The authors use the theorems of Abel and \textit{U. Zannier} [Am. J. Math. 141, No. 1, 1--40 (2019; Zbl 1422.11156)] for a polynomial of even degree \(D(t)\) with complex coefficients, to the sequence of the degrees of the partial quotients in the continued fraction expansion of \(\sqrt{D(t)}\) is eventually periodic, even when the expansion itself is not. The authors discuss the case in which the curve \(y^2=D(t)\) has genus 0, establishing explicit geometric conditions corresponding to the appearance of partial quotients of certain degrees in the continued fraction expansion.They also express that there are non-trivial polynomials \(D(t)\) with non-periodic expansions such that infinitely many partial quotients have degree greater than one. The method is new and the work done by the authors in this paper is highly appreciable and useful for further research.