I've already seen methods of solving equations of the form $x^p+y^q=z^r$ using the fact that $$2^a3^b+2^{a+1}3^b=2^a3^{b+1}.$$
Similarly, can we let, for $a,j,A\in \mathbb{Z}$, $$\Omega =\sum_{j_1,\dots,j_k}a_{j_1,...,j_k}\prod_{1\leq i\leq k} A_i^{j_i},$$ and consider $$\sum_i B^{p_i}_i=\beta^q$$ (for $\beta,B,p,q$ in the same field) comparing this to $$\Omega =\prod A_i^{k_i}?$$ and $k\in\mathbb{Z}$. I'm not sure I stated this correctly or fully: for example, could we compare
$$\eta^a(1+\eta+A\eta^2)^b+\eta^{a+1}(1+\eta+A\eta^2)^b+A\eta^{a+2}(1+\eta+A\eta^2)^b=\eta^a(1+\eta+A\eta^2)^{b+1}$$ to $$x^p+y^q+Az^r=t^s?$$ Over finite fields? Are there simpler methods?