• Computing $n$-th roots in $\rm{SL_{2}}$ and Fibonacci polynomials

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    • Keywords


      $SL_{2}$; $n$-th roots; Fibonacci polynomials

    • Abstract


      Let $k$ be a field of characteristic $\neq$2. In this paper, we study squares, cubesand their products in split and anisotropic groups of type $A_{1}$. In the split case,we show that computing $n$-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 $n$-th powers, and conjugacy classes which are $n$-th powers, in $SL_{2}(\mathbb{F}_{q})$ when $n$ is a prime or $n = 4$. We also extend the already known Waring type result for $SL_{2}(\mathbb{F}_{q})$, that every element of $SL_{2}(\mathbb{F}_{q})$ is a product of two squares, to $SL_{2}(k)$ for an arbitrary $k$. For anisotropic groups of type $A_{1}$, namely $SL_{1}(Q)$ where $Q$ is a quaternion division algebra, we prove that when 2 is a square in $k$, every element of $SL_{1}(Q)$ is a product of two squares if and only if $−1$ is a square in $SL_{1}(Q)$.

    • Author Affiliations



      1. Indian Institute of Science Education and Research Mohali, Knowledge City, Sector 81, Mohali 140 306, India
      2. Indian Institute of Science Education and Research Pune, Dr. Homi Bhabha Road, Pashan, Pune 411 008, India
    • Dates

  • Proceedings – Mathematical Sciences | News

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