A Positivstellensatz on the Matrix Algebra of Finitely Generated Free Group

TL;DR

This paper establishes a Positivstellensatz for matrix algebras over free groups, linking positivity of symmetric polynomials with sums of Hermitian squares via unitary representations.

Contribution

It provides a new proof connecting positivity conditions of polynomials in free group variables to positive semidefinite matrices through real algebraic geometry.

Findings

A polynomial is a sum of Hermitian squares iff it remains positive under all finite-dimensional unitary representations.

The paper extends Positivstellensatz concepts to noncommutative free group algebras.

New proof techniques relate algebraic positivity to operator-theoretic positivity.

Abstract

Positivstellens{\"a}tze are a group of theorems on the positivity of involution algebras over $\mathbb{R}$ or $\mathbb{C}$. One of the most well-known Positivstellensatz is the solution to Hilbert's 17th problem given by E. Artin, which asserts that a real polynomial in $n$ commutative variables is nonnegative on real affine space if and only if it is a sum of fractional squares. Let $m$ and $n$ be two positive integers. For the free group $F_n$ generated by $n$ letters, and a symmetric polynomial $b$ with variables in $F_n$ and with $n$-by-$n$ complex matrices coefficients, we use real algebraic geometry to give a new proof showing that $b$ is a sum of Hermitian squares if and only if $b$ is mapped to a positive semidefinite matrix under any finitely dimensional unitary representation of $F_n$.