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

Amit Kulshrestha, Anupam Singh

Published 2017-10-10Version 1

Let $k$ be a field of characteristic $\neq 2$. In this paper, we show that computing $n^{\rm th}$ root of an element of the group $SL_2(k)$ is equivalent to finding solutions of certain polynomial equations over the base field $k$. These polynomials are in two variables, and their description involves generalised Fibonacci polynomials. As an application, we prove some results on surjectivity of word maps over $SL_2(k)$. We prove that the word maps $X_1^2X_2^2$ and $X_1^4X_2^4X_3^4$ are surjective on $SL_2(k)$ and, with additional assumption that characteristic $\neq 3$, the word map $X_1^3X_2^3$ is surjective. Further, over finite field $\mathbb F_q$, $q$ odd, we show that the proportion of squares and, similarly, the proportion of conjugacy classes which are square in $SL_2(\mathbb F_q)$, is asymptotically $\frac{1}{2}$. More generally, for $n\geq 3$, a prime not dividing $q$ but dividing the order of $SL_2(\mathbb F_q)$, we show that the proportion of $n^{th}$ powers, and, similarly the proportion of conjugacy classes which are $n^{th}$ power, in $SL_2(\mathbb F_q)$, is asymptotically $\frac{n+1}{2n}$. This gives an alternate proof of the result that $X_1^nX_2^n$ is surjective on $SL_2(\mathbb F_q)$ except for $n=q=3$. We further conclude that for $m,n$ odd primes the word map $X_1^mX_2^n$ is surjective on $SL_2(\mathbb F_q)$ except when $m=n=q=3$.