Hereditary NC Polynomials & Positivity Domains

Abstract

Noncommutative polynomials, elements of the free algebra generated by several noncommuting variables $x_1,...,x_g,x_1^{\ast },...,x_g^{\ast }$, are of importance in noncommutative algebra, free analysis and real algebraic geometry. We allow matrix coefficients and view these noncommutative polynomials as functions in several size-free matrix variables. The free locus of a noncommutative polynomial $f$ is the sequence of hypersurfaces ${\mathcal{Z}}_n(f)=\{X\in M_n(\mathbb{C}{)}^g:\det f(X)=0\}$. When $f$ is Hermitian, the positivity domain of $f$ is the sequence of semialgebraic sets ${\mathcal{D}}_n(f)=\{X\in M_n(\mathbb{C}{)}^g:f(X)\text{ is positive semidefinite}\}$. We show for well-behaved cases that the Euclidean boundary of the positivity domain is Zariski dense in the free locus. This leads to a certificate for equality of positivity domains of coefficient-irreducible unsignatured atomic quadratics. Finally, we describe the geometric structure of positivity domains of hereditary quadratic polynomials through $\Gamma$-convexity.