Let $k$ be a number field. Let $M_0,M_1,\ldots,M_r \in k[x_1,\ldots,x_n]$ be monomials, with $r,n \geq 2$. Assume, for some constants $c_1,\ldots,c_r$, binomials $$B_1=M_0+c_1M_1$$ $$\cdots$$ $$B_r=M_0+c_rM_r$$ have a common divisor $D$ which is not a monomial. I would like to know if there is something we can say about $c_1,\ldots,c_r$.
A possible relation I can see so far is that $c_1,\ldots,c_r$ might be multiplicatively dependent. For example, $$x^4-16\text{ and }x^4-4x^2 \text{ and } x^4-2x^3,$$ they share a common divisor $x-2$ and $c_i's$ are multiplicatively dependent.
But I am not sure how to prove it. Or maybe there is a counterexample?