I am looking for a formula like $a^n-b^n=(a-b)(a^{n-1}+a^{n-2}b+\cdots+b^{n-1})$, but for multivariable monomials.
More specifically, I want to write $a_1^{n_1}a_2^{n_2}\cdots a_k^{n_k} - b_1^{n_1}b_2^{n_2}\cdots b_k^{n_k}$ "in terms of" $a_1-b_1$, $a_2-b_2$, ... $a_k-b_k$ somehow. I don't know exactly what I'm looking for, but it should be something like this.
I am trying to upgrade a result about monomials of one variable to homogeneous polynomials of several variables, and I think this would be a very helpful tool.
Thanks in advance for anything related. I have tried to google it but haven't had any luck.