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: ${\displaystyle (xy-yx)^{2}}$ is a central polynomial for 2-by-2-matrices. Indeed, by the Cayley–Hamilton theorem, one has that ${\displaystyle (xy-yx)^{2}=-\det(xy-yx)I}$ 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. Zbl 0714.16001.
• Artin, Michael (1999). "Noncommutative Rings" (PDF). V. 4.CS1 maint: location (link)