Central polynomial

Remarks

Clause 1 says that all values of $f$ lie in the center $k I$ of $M_n(k)$.

Example: $f = (x_1 x_2 - x_2 x_1)^2$ is a central polynomial for $M_2(k)$. For $a, b \in M_2(k)$ the matrix $c = ab - ba$ has trace $0$, so the Cayley–Hamilton theorem gives $c^2 = -\det(c) \, I$, a scalar matrix. For $a = e_{12}$ and $b = e_{21}$ one has $c = e_{11} - e_{22}$ and $c^2 = I \ne 0$, so clause 2 holds.

Non-example: $g = x_1 x_2 - x_2 x_1$ fails clause 1 for $M_2(k)$, since $g(e_{12}, e_{21}) = e_{11} - e_{22}$ is not scalar. Non-example: $s_4$, the standard polynomial of degree $4$, satisfies clause 1 for $M_2(k)$ only because all its values are $0$, so it fails clause 2.

For $n = 1$ every polynomial with a nonzero value on $k$ is central, so the notion is of interest for $n \ge 2$. Central polynomials exist for every $n \ge 1$, a result of Formanek and of Razmyslov.