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: $(xy-yx{)}^2$ is a central polynomial for 2-by-2-matrices. Indeed, by the Cayley–Hamilton theorem, one has that $(xy-yx{)}^2=-\det (xy-yx)I$ for any 2-by-2-matrices x and y.

See also

• Generic matrix ring

References

• Formanek, Edward (1991). The polynomial identities and invariants of n×n matrices. Regional Conference Series in Mathematics. 78. Providence, RI: American Mathematical Society. ISBN 0-8218-0730-7. 
• Artin, Michael (1999). "Noncommutative Rings". V. 4.. http://math.mit.edu/~etingof/artinnotes.pdf.