Central polynomial

In algebra, a central polynomial for "n"-by-"n" matrices is a polynomial in non-commuting variables that is non-constant but yields a scalar matrix whenever it is evaluated at "n"-by-"n" matrices. That such polynomials exist for any square matrices was discovered in 1970 independently by Formanek and Razmyslov. The term "central" is because the evaluation of a central polynomial has the image lying in the center of the matrix ring over any commutative ring. The notion has an application to the theory of polynomial identity rings.
Example: formula_1 is a central polynomial for 2-by-2-matrices. Indeed, by the Cayley–Hamilton theorem, one has that formula_2 for any 2-by-2-matrices "x" and "y".