Is there any known generalization of Jacobian conjecture which gives condition for $k[f_1, \ldots, f_m] = k[g_1, \ldots, g_m]$ where all $f_i$ and $g_i$ are functions over $x_1, \ldots, x_n$? Note that possibly, $n \neq m$.
Is there anything known even if we assume $n = m$?
One possibility is that the determinant of Jacobian for $f$ and $g$ (as in Jacobian conjecture) is same w.r.t. all $m$-tuples of $x_i$'s. This is stronger than Jacobian conjecture, but I cannot prove the other way round.