A local-global principle for linear dependence of noncommutative polynomials

TL;DR

This paper presents an alternative proof for a local-global principle in noncommutative polynomial linear dependence, extending the result to broader algebraic contexts and providing bounds on matrix sizes for testing dependence.

Contribution

It offers a new proof based on polynomial identities, generalizes to directional dependence and arbitrary characteristic fields, and establishes bounds on matrix sizes for testing linear dependence.

Findings

Finite local linear dependence implies global linear dependence.

The proof extends to evaluations in general algebras over arbitrary fields.

Bounds on matrix sizes needed for dependence testing are established.

Abstract

A set of polynomials in noncommuting variables is called locally linearly dependent if their evaluations at tuples of matrices are always linearly dependent. By a theorem of Camino, Helton, Skelton and Ye, a finite locally linearly dependent set of polynomials is linearly dependent. In this short note an alternative proof based on the theory of polynomial identities is given. The method of the proof yields generalizations to directional local linear dependence and evaluations in general algebras over fields of arbitrary characteristic. A main feature of the proof is that it makes it possible to deduce bounds on the size of the matrices where the (directional) local linear dependence needs to be tested in order to establish linear dependence.