Let $F$ be an infinite field and let $f \in F[x\_{11},x\_{12},...,x\_{nn}]$ be an **arbitrary** polynomial in $n^2$ variables. Consider the function $\phi : M\_n(F)\longrightarrow F$ defined by $\phi((a\_{ij})) = f(a\_{11},a\_{12}, ..., a\_{nn})$ and suppose that $\phi(I\_n) = 1$ and $\phi(AB)= \phi(A)\phi(B)$, for any $A,B \in M\_n(F)$. Is it true that $f$ is equal to some power of the determinant (considered as a polynomial of $n^2$ valiables $x\_{ij}$). 

**Comment:** When $f$ is a homogeneous polynomial then the problem is known to be true but I have no idea for the general case.