Norm-constrained determinantal representations of polynomials

TL;DR

This paper constructs determinantal representations for multivariable polynomials using matrices with norm constraints, linking these representations to von Neumann inequalities, Agler denominators, and stability in multivariable operator theory.

Contribution

It introduces a method to represent multivariable polynomials as determinants with norm-constrained matrices, connecting to key concepts in multivariable operator theory.

Findings

Determinantal representations exist for polynomials with certain linear relations among minors.

Norm constraints on the matrix K relate to von Neumann inequality and stability.

Contractive K matrices correspond to rational inner functions in the Schur–Agler class.

Abstract

For every multivariable polynomial $p$, with $p(0)=1$, we construct a determinantal representation $$p=\det (I - K Z),$$ where $Z$ is a diagonal matrix with coordinate variables on the diagonal and $K$ is a complex square matrix. Such a representation is equivalent to the existence of $K$ whose principal minors satisfy certain linear relations. When norm constraints on $K$ are imposed, we give connections to the multivariable von Neumann inequality, Agler denominators, and stability. We show that if a multivariable polynomial $q$, $q(0)=0,$ satisfies the von Neumann inequality, then $1-q$ admits a determinantal representation with $K$ a contraction. On the other hand, every determinantal representation with a contractive $K$ gives rise to a rational inner function in the Schur--Agler class.