Let $k$ be a field. Let $F$ be a polynomial in $k[x_{1},\ldots,x_{n},y_{1},\ldots,y_{n}]$ with the following "symmetric" property (S):
If we write $F$ explicitly as $\displaystyle F=\sum_{i} c_{i} (\prod_{j_{i}} x_{j_{i}}^{p_{j_{i}}}\prod_{k_{i}}y_{k_{i}}^{q_{k_{i}}})$, $c_{i}$ constant and $p_{j_{i}},q_{k_{i}}$ positive integers. Then $p_{j_{i}}=q_{k_{i}}$ if $j_{i}=k_{i}$ (in other word, this means that inside each summand, $x$'s and $y$'s with same indices appear in pairs).
I am guessing if I have a nontrivial divisor of this $F$, it should also satisfy property (S), but not quite sure. Also failed to find any counterexample.
Any comments, hint towards proof or a counterexample is welcome. Thanks!