Images of Multilinear Polynomials in the Algebra of Finitary Matrices Contain Trace Zero Matrices
Daniel Vitas

TL;DR
The paper proves that the image of any nonzero multilinear polynomial over an infinite field includes all trace zero matrices in finitary matrix algebras, revealing a broad algebraic property.
Contribution
It establishes that for any positive integer, the image of a multilinear polynomial on finitary matrices encompasses all trace zero matrices, a novel generalization in matrix algebra.
Findings
Image of f contains all trace zero matrices in finitary algebra.
For each dimension, there exists a size s where the image includes all trace zero matrices.
The result applies to all finitary matrices over infinite fields.
Abstract
Let be an infinite field and let be a nonzero multilinear polynomial with coefficients in . We prove that for every positive integer there exists a positive integer such that , the image of in , contains all trace zero matrices. In particular, the image of in the algebra of all finitary matrices contains all trace zero finitary matrices.
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Taxonomy
TopicsAdvanced Topics in Algebra · Matrix Theory and Algorithms · Advanced Differential Equations and Dynamical Systems
