15.63 (2002)

Let $F_n$ be the free group of finite rank $n$ on the free generators $x_1, \dots, x_n$. An element $u \in F_n$ is called positive if $u$ belongs to the semigroup generated by the $x_i$. An element $u \in F_n$ is called potentially positive if $\alpha(u)$ is positive for some automorphism $\alpha$ of $F_n$.
$\qquad$ a) Is the property of an element to be potentially positive algorithmically recognizable?
$\qquad$ b) Finally, $u \in F_n$ is called stably potentially positive if it is potentially positive as an element of $F_m$ for some $m \geqslant n$. Are there stably potentially positive elements that are not potentially positive?