Indefinite determinantal representations versus nonsingularities on the noncommutative d-torus

TL;DR

This paper explores the relationship between indefinite determinantal representations of multivariable polynomials and their nonsingularity on the noncommutative d-torus, linking matrix structure to spectral properties of matrix tuples.

Contribution

It establishes a characterization connecting the structure of the matrix in a determinantal representation to the polynomial's extension's nonsingularity on the noncommutative d-torus.

Findings

Matrix $K$ is similar to a strictly $J$-contractive matrix if and only if the polynomial extension has no roots on the noncommutative d-torus.

Extension of polynomial to matrix tuples preserves nonsingularity if $K$ has the specified structure.

Provides a criterion for nonsingularity of multivariable polynomials on the noncommutative torus based on determinantal representations.

Abstract

We show that for a multivariable polynomial $p(z)=p(z_1, \ldots , z_d)$ with a determinantal representation $$ p(z) = p(0) \det (I_n- K (\oplus_{j=1}^d z_j I_{n_j}))$$ the matrix $K$ is structurally similar to a strictly $J$-contractive matrix for some diagonal signature matrix $J$ if and only if the extension of $p(z)$ to a polynomial in $d$-tuples of matrices of arbitrary size given by \[ p(U_1, \ldots , U_d) = p(0,\ldots,0) \det (I_n\otimes I_m- (K\otimes I_m) (\oplus_{j=1}^d I_{n_j}\otimes U_j)), \] where $U_1,\ldots , U_d \in {\mathbb C}^{m \times m}$, $m\in {\mathbb N}$, does not have roots on the noncommutative $d$-torus consisting of $d$-tuples $(U_1, \ldots , U_d)$ of unitary matrices of arbitrary size.