Continued fractions in function fields: polynomial analogues of McMullen's and Zaremba's conjectures

TL;DR

This paper explores polynomial analogues of McMullen's and Zaremba's conjectures in function fields, proving their validity over various fields and connecting continued fractions to algebraic geometry.

Contribution

It provides new proofs and extends the validity of polynomial analogues of these conjectures over infinite, finite, and algebraic extension fields, linking continued fractions to Jacobians.

Findings

Polynomial analogue of Zaremba's conjecture holds over infinite fields.

Polynomial analogue of McMullen's conjecture holds over uncountable fields and number fields.

Connection established between continued fractions and Jacobians of hyperelliptic curves.

Abstract

We examine the polynomial analogues of McMullen's and Zaremba's conjectures on continued fractions with bounded partial quotients. It has already been proved by Blackburn that if the base field is infinite, then the polynomial analogue of Zaremba's conjecture holds; we will prove this again with a different method and examine some known results for finite base fields. Translating to the polynomial setting a result of Mercat, we will prove that the polynomial analogue of McMullen's conjecture holds over infinite algebraic extensions of finite fields and that, over finite fields, it would be a consequence of the polynomial analogue of Zaremba's conjecture. We will then prove that the polynomial analogue of McMullen's conjecture holds over uncountable base fields, over $\overline{\mathbb Q}$ (thanks to the theory of reduction of a formal Laurent series modulo a prime) and over number fields. For this purpose, we will examine the connection between the continued fractions of polynomial multiples of $\sqrt D$ and pullbacks of generalized Jacobians of the hyperelliptic curve $U^2=D(T)$.