Solving systems of equations of raising-to-powers type

We address special cases of the analogues of the exponential algebraic closedness conjecture relative to the exponential maps of semiabelian varieties and to the modular j function. In particular, we show that the graph of the exponential of an abelian variety intersects products of rotund varieties in which the subvariety of the domain is a sufficiently generic linear subspace, and that the graph of j intersects products of free broad varieties in which the subvariety of the domain is a Möbius subvariety.