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Computing $n^{\rm th}$ roots in $SL_2(k)$ and Fibonacci polynomials. (arXiv:1710.03432v1 [math.GR])
来源于:arXiv
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,
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