Computing $n^{\rm th}$ roots in $SL_2$ and Fibonacci polynomials

TL;DR

This paper investigates the computation of $n^{ m th}$ roots in groups of type $A_1$, using polynomial equations involving Fibonacci polynomials, and extends known results about expressing elements as products of squares in these groups.

Contribution

It provides a polynomial equation framework for $n^{ m th}$ roots in split groups and extends Waring type results to arbitrary fields, also characterizing products of squares in anisotropic groups.

Findings

Computing $n^{ m th}$ roots reduces to solving polynomial equations with Fibonacci polynomials.

Asymptotic proportions of $n^{ m th}$ powers in ${ m SL}_2( ext{finite field})$ are obtained.

Every element of ${ m SL}_2(k)$ is a product of two squares over arbitrary fields under certain conditions.

Abstract

Let $k$ be a field of characteristic $\neq 2$. In this paper we study squares, cubes and their products in split and anisotropic groups of type $A_1$. In split case, we show that computing $n^{\rm th}$ roots is equivalent to finding solutions of certain polynomial equations in at most two variables over the base field $k$. The description of these polynomials involves generalised Fibonacci polynomials. Using this we obtain asymptotic proportions of $n^{\rm th}$ powers, and conjugacy classes which are $n^{\rm th}$ powers, in ${\rm SL}_2(\mathbb F_q)$ when $n$ is a prime or $n = 4$. We also extend already known Waring type result for ${\rm SL}_2(\mathbb F_q)$, that every element of ${\rm SL}_2(\mathbb F_q)$ is a product of two squares, to ${\rm SL}_2(k)$ for an arbitrary $k$. For anisotropic groups of type $A_1$, namely ${\rm SL}_1(Q)$ where $Q$ is a quaternion division algebra, we prove that when $2$ is a square in $k$, every element of ${\rm SL}_1(Q)$ is a product of two squares if and only if $-1$ is a square in ${\rm SL}_1(Q)$.