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Preserving zeros of a polynomial. (English) Zbl 1183.15026

The article studies the following open problem:
Let \(p(x_1,\dots,x_k)\) be a polynomial over a field \(F\) in noncommuting indeterminates \(x_1,\dots,x_k\) of degree \(deg \, p > 1\) and \(\Phi : M_n(F) \to M_n(F)\) be a linear map on the matrix algebra \(M_n(F)\) of all \(n \times n\) matrices over the field \(F\). Suppose that \(p(A_1,\dots,A_k)=0\) implies \(p(\Phi(A_1),\dots,\Phi(A_k))=0\). Is it possible to describe such \(\Phi\)?
The paper tackles this open problem even in the case of nonlinear \(\Phi\). For the homogeneous multilinear polynomials with nonzero sum of coefficients, the problem is completely solved over algebraically closed fields. For the case of zero sum coefficients, the problem is solved for certain rather large classes of polynomials of general type.
All results are of nonlinear nature so that no additivity of a transformation is needed. Further, the transformation is not necessarily required to be invertible; it is only assumed to be surjective. Under some conditions, even surjectivity is no longer needed. The technique is characteristic free and allows one to work without restrictions on the number of variables of a polynomial. But, constraints must be imposed because there exist polynomial identities in \(M_n(F)\).

MSC:

15A86 Linear preserver problems
12E05 Polynomials in general fields (irreducibility, etc.)
15A30 Algebraic systems of matrices
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