Matrix evaluations of noncommutative rational functions and Waring problems
arxiv(2024)
摘要
Let r be a nonconstant noncommutative rational function in m variables
over an algebraically closed field K of characteristic 0. We show that for
n large enough, there exists an X∈ M_n(K)^m such that r(X) has n
distinct and nonzero eigenvalues. This result is used to study the linear and
multiplicative Waring problems for matrix algebras. Concerning the linear
problem, we show that for n large enough, every matrix in sl_n(K) can be
written as r(Y)-r(Z) for some Y,Z∈ M_n(K)^m. We also discuss variations
of this result for the case where r is a noncommutative polynomial.
Concerning the multiplicative problem, we show, among other results, that if
f and g are nonconstant polynomials, then, for n large enough, every
nonscalar matrix in GL_n(K) can be written as f(Y)g(Z) for some Y,Z∈
M_n(K)^m.
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