MR3038546, Brešar, Matej; Klep, Igor A local-global principle for linear dependence of noncommutative polynomials. Israel J. Math. 193 (2013), no. 1, 71–82. (Reviewer: Daniela La Mattina) 16R99

Abstract

Let F be a eld of characteristic zero and FhXi the free associative algebra on X = fX1;X2; : : : g over F; i.e., the algebra of polynomials in the non-commuting variables Xi 2 X. A set of polynomials in FhXi is called locally linearly dependent if their evaluations at tuples of matrices are always linearly dependent. In [Integral Equations Operator Theory 46 (2003), no. 4, 399{454; MR1997979 (2004f:90102)], J. F. Camino et al., in the setting of free analysis, motivated by systems engineering, proved that a nite locally linearly dependent set of polynomials is linearly dependent. In this paper the authors give an alternative algebraic proof of this result based on the theory of polynomial identities. As such it applies not only to matrix algebras but also to evaluations in general algebras over elds of arbitrary characteristic. Moreover, it makes it possible to extract bounds on the size of the matrices where the local linear dependence needs to be checked in order to establish linear dependence.