simpl.tex

\begin{frame}[allowframebreaks]{Roots of $f^n(x)$}

  % TODO show root tree?

  $f(x) = (x-\gamma)^2+\gamma+m$

  \begin{itemize}
    \item The roots of $f(x)$ are $\gamma\pm\sqrt{-m-\gamma}$
    \item If $\alpha$ is a root of $f^n(x)$, then $\gamma\pm\sqrt{\alpha-m-\gamma}$ are roots of $f^{n+1}(x)$
  \end{itemize}

  \begin{obs}
    For $n>0$, the roots of $f^n(x)$ are, with $n$ radicals:
    $$\gamma\pm\sqrt{-m\pm\sqrt{-m\pm\sqrt{-m\pm\ldots\sqrt{-m-\gamma}}}}$$
  \end{obs}

  \framebreak

  \begin{obs}
    For $n>0$, the roots of $f^n(x)$ are, with $n$ radicals:
    $$\gamma\pm\sqrt{-m\pm\sqrt{-m\pm\sqrt{-m\pm\ldots\sqrt{-m-\gamma}}}}$$
  \end{obs}

  For notational convenience, define $\beta : \Sigma^* \rightarrow \CC$ where
  \begin{align*}
    \beta_{\epsilon} &= -\gamma \\
    \beta_{0s} &= \sqrt{-m+\beta_s} \\ 
    \beta_{1s} &= -\sqrt{-m+\beta_s}
  \end{align*}

  For $n>0$, the roots of $f^n(x)$ are exactly $\{\;\gamma+\beta_s\mid s\in\Sigma^n\;\}$.

\end{frame}