I need four non-trivial polynomials $P(x)$, $Q(y)$, $R(z)$ and $S(w)$ such that $$P(x)Q(y)R(z)S(w)=\sum_{i}a_i b_i c_i d_i x^i y^i z^i w^i+ \sum_{j\neq k\neq l\neq m}e_j f_k g_l h_m x^j y^k z^l w^m $$ where $i\geq 0$ and $j,k,l,m\geq 1$
So in this example in the RHS we want only terms of the type $x^2 y^2 z^2w^2$, $x^5y^6z^8w^3$ etc. and no terms of the type $x^2 y^2 z^5 w^5$, $x^3 y^4 $, $x^6 y^6z^6 w^7$ etc.
Any help would be highly appreciated.
Edit I tried for the above product to have all the powers distinct as $$ x^{n+1}(1-x)^n y^{2n+2}(1-y)^n (1-z)^n w^{3n+3}(1-w)^{n}= \sum_{j\neq k\neq l\neq m}e_j f_k g_l h_m x^j y^k z^l w^m $$ How do I get a polynomial with all the powers as same? In the above polynomial also can we get by some manipulations,some terms which have all the powers same?
I don't think the above example works.