Exponential-polynomial equations and dynamical return sets

We show that for each finite sequence of algebraic integers $\alpha_1,...,\alpha_n$ and polynomials $P_1(x_1,...,x_n;y_1,...,y_n),..., P_r(x_1,...,x_n;y_1,...,y_n)$ with algebraic integer coefficients, there are a natural number $N$, $n$ commuting endomorphisms $\Phi_i:\Gm^N \to \Gm^N$ of the $N^\text{th}$ Cartesian power of the multiplicative group, a point $P \in \Gm^N(\QQ)$, and an algebraic subgroup $G \leq \Gm^N$ so that the return set $\{(\ell_1,...,\ell_n) \in \NN^n : \Phi_1^{\circ \ell_1} \circ... \circ \Phi_n^{\circ \ell_n}(P) \in G(\QQ) \}$ is identical to the set of solutions to the given exponential-polynomial equation: $\{(\ell_1,...,\ell_n) \in \NN^n : P_1(\ell_1,...,\ell_n;\alpha_1^{\ell_1},...,\alpha_n^{\ell_n}) = ... = P_r(\ell_1,...,\ell_n;\alpha_1^{\ell_1},...,\alpha_n^{\ell_n}) = 0 \}$.