Trace Minimization and Roots in $\mathrm{P}\mathrm{S}\mathrm{L}(2,\mathbb{R})$

Suppose that $A,B\in \mathrm{P}\mathrm{S}\mathrm{L}(2,\mathbb{R})$ generate a discrete and free group of rank 2, and let $m,n\ge 1$. We consider subgroups $⟨R,S⟩$ of $\mathrm{P}\mathrm{S}\mathrm{L}(2,\mathbb{R})$ generated by roots of $A$ and $B$, i.e., by elements such that $R^m=A$ and $S^n=B$. Depending on whether the commutator trace $\tau =\mathrm{t}\mathrm{r}([A,B])$ is larger or smaller than 2, we describe necessary and sufficient conditions for $⟨R,S⟩$ to be discrete and free of rank 2. For $\tau \le -2$, this can be checked with an explicit formula. For $\tau >2$, one has to use the Trace Minimization Algorithm. Besides an explicit formulation of this algorithm, we prove new formulas for the powers and roots of elements of $\mathrm{P}\mathrm{S}\mathrm{L}(2,\mathbb{R})$, their traces and their commutator traces. The case of positive rational exponents $m,n$ is treated, as well.