TY - JOUR

T1 - 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

AU - La Mattina, Daniela

PY - 2014

Y1 - 2014

N2 - 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 variablesXi 2 X. A set of polynomials in FhXi is called locally linearly dependent if theirevaluations at tuples of matrices are always linearly dependent. In [Integral EquationsOperator Theory 46 (2003), no. 4, 399{454; MR1997979 (2004f:90102)], J. F. Caminoet 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 onthe theory of polynomial identities. As such it applies not only to matrix algebras butalso 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 lineardependence needs to be checked in order to establish linear dependence.

AB - 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 variablesXi 2 X. A set of polynomials in FhXi is called locally linearly dependent if theirevaluations at tuples of matrices are always linearly dependent. In [Integral EquationsOperator Theory 46 (2003), no. 4, 399{454; MR1997979 (2004f:90102)], J. F. Caminoet 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 onthe theory of polynomial identities. As such it applies not only to matrix algebras butalso 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 lineardependence needs to be checked in order to establish linear dependence.

UR - http://hdl.handle.net/10447/103318

UR - http://www.ams.org/mathscinet/search/publdoc.html?pg1=RVRI&pg3=authreviews&s1=734661&vfpref=html&r=1&mx-pid=3038546

M3 - Review article

VL - MR3038546

JO - MATHEMATICAL REVIEWS

JF - MATHEMATICAL REVIEWS

SN - 0025-5629

ER -