On continued fraction expansions of quadratic irrationals in positive characteristic

TL;DR

This paper investigates the continued fraction expansions of quadratic irrationals over finite fields, revealing distinct asymptotic behaviors in positive characteristic compared to zero characteristic.

Contribution

It introduces new asymptotic results for degrees of coefficients in continued fractions of quadratic irrationals over finite fields, contrasting with classical zero characteristic cases.

Findings

Degrees of coefficients grow differently in positive characteristic

Asymptotic properties differ sharply from zero characteristic cases

Provides new insights into continued fractions over finite fields

Abstract

Let $P$ be a prime polynomial in the variable $Y$ over a finite field and let $f$ be a quadratic irrational in the field of formal Laurant series in the variable $Y^{-1}$. We study the asymptotic properties of the degrees of the coefficients of the continued fraction expansion of quadratic irrationals such as $P^nf$ and prove results that are in sharp contrast to the analogue situation in zero characteristic.