$(x^2+xy+y^2)(z^2+zw+w^2)=(xz-yw)^2+(xz-yw)[wx+y(z+w)]+[wx+y(z+w)]^2$

\begin{vmatrix}
x& y\cr
-y & x+y
\end{vmatrix}

Let $$f(x_1,x_2,\dotsc,x_n)$$ be a homogeneous polynomial. Let

$S=\{f(a_1,a_2,\dotsc,a_n)\mid a_1,a_2,\dotsc,a_n \in\Bbb Z\}.$

If $$S$$ satisfies the following condition: for all $$m,n\in S$$, we have $$mn\in S$$. Can we determine all the homogeneous polynomials $$f$$?

For example, $$x^n(n\in\Bbb N),x^2+n y^2(n\in\Bbb Z), x^2+xy+y^2,x^3+y^3+z^3-3xyz$$, and $$x^2+y^2+z^2+w^2$$ are all appropriate examples.

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