On the image of polynomials evaluated on incidence algebras: a counter-example and a solution

In this paper, we investigate the subset obtained by evaluations of a fixed multilinear polynomial on a given algebra. We provide an example of a multilinear polynomial, whose image is not a vector subspace; namely, the product of two commutators need not to be a subspace whenever evaluated on certa...

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Bibliographic Details
Published in:arXiv.org 2019-04
Main Authors: Santulo, Ednei A, Yasumura, Felipe Y
Format: Article
Language:English
Subjects:
Algebra
Chief executive officers
Commutators
Fields (mathematics)
Incidence
Matrices (mathematics)
Polynomials
Set theory
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Summary:In this paper, we investigate the subset obtained by evaluations of a fixed multilinear polynomial on a given algebra. We provide an example of a multilinear polynomial, whose image is not a vector subspace; namely, the product of two commutators need not to be a subspace whenever evaluated on certain subalgebras of upper triangular matrices (the so-called incidence algebras). In the last part of the paper, given that the field is infinite, we reduce the problem of the description of the image of a polynomial evaluated on an incidence algebra to the study of evaluations of a certain family of polynomials on its Jacobson radical. In particular, we are able to describe the image of multilinear polynomials evaluated on the algebra of upper triangular matrices.
ISSN:2331-8422