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arXiv · 1902.08116

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

Ednei A. Santulo Jr.·Felipe Y. Yasumura
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Abstract

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.

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Ednei A. Santulo Jr., Felipe Y. Yasumura. 2019-02-21. On the image of polynomials evaluated on incidence algebras: a counter-example and a solution. https://arxiv.org/abs/1902.08116

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