Preserving Zeros of a Polynomial

Abstract

We study non-linear surjective mappings on subsets of M n (F), which preserve the zeros of some fixed polynomials in noncommuting variables.Mathematics subject classification (2000): 15A99, 16W99.

Linear K-Power Preservers and Trace of Power-Product Preservers

“…We may further explore linear f -preservers for a multivariable function fx 1 , … , x r ðÞ , that is, operator ψ satisfying ψ fA 1 , … , A r ðÞ ðÞ ¼ f ψ A 1 ðÞ , … , ψ A r ðÞ ðÞ . The corresponding annihilator preserver problem has been studied in some special cases, for example, on M n for homogeneous multilinear polynomials by A. E. Guterman and B. Kuzma in [25].…”

“…Let n ¼ k þ m, k, m ≥ 2,and consider the operator ψ on the subspaceW ¼ M k ⊕M m of M n defined by ψ A⊕B ðÞ ¼ A⊕B t for A ∈ M k and B ∈ M m : Then ψ A k ÀÁ ¼ ψ A ðÞ k for all A ∈ W and k ∈  þ , but ψ is not of the form(25) or (26). We now generalize Theorem 3.1 to include negative integers k and to assume the kpower preserving condition ψ A k ÀÁ ¼ ψ A ðÞ k only on matrices nearby the identity.…”

Maps preserving matrix pairs with zero Jordan product

“…zeros of a Lie product (X, Y )°XY À YX, were classified by Watkins [31], while general ones were given by Sˇemrl [28]. Maps on matrices that preserve zeros of a fixed homogeneous multilinear polynomial in k non-commuting variables were studied by Guterman and Kuzma [10]. We refer to [7] for linear preservers of zero product on quite general algebras, and to [34] for additive preservers of zeros of Jordan product on certain operator algebras.…”

Maps preserving zeros of a polynomial

“…So far not much is known for general polynomials. For them the problem was explicitly posed by Chebotar et al [9] for the matrix algebra A = M n (F ), and some partial solutions were obtained in two recent papers: [13] considers, in particular, the case where the sum of coefficients of f is a nonzero scalar (without assuming the linearity of φ), and [10] handles Lie polynomials of degree at most 4. Let us also mention a related, yet considerably simpler, problem of describing linear maps that preserve all values of f , i.e., φ(f (a 1 , .…”