2021/08/14 by Francesco Ballini, Ballini, Francesco, Francesco Veneziano +1
Chemistry · Computer Science · Mathematics · #Advanced Differential Equations and Dynamical Systems #Chromatography in Natural Products #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2108.06560
openalex publication_date 2021/08/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
It is possible to define a continued fraction expansion of elements in a function field of a curve by expanding as a Laurent series in a local parameter. Considering the square root of a polynomial √(D(t)) leads to an interesting theory related to polynomial Pell equations. Unlike the classical Pell equation, the corresponding polynomial equation is not always solvable and its solvability is related to arithmetic conditions on the Jacobian (or generalized Jacobian) of the curve defined by y2=D(t). In this setting, it has been shown by Zannier in \citezannier that the sequence of the degrees of the partial quotients of the continued fraction expansion of √(D(t)) is always periodic, even when the expansion itself is not. In this article we work out in detail the case in which the curve y2=D(t) has genus 0, 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 D(t) with non-periodic expansions such that infinitely many partial quotients have degree greater than one.