Indefinite determinantal representations versus nonsingularities on the noncommutative d-torus

Abstract

We show that for a multivariable polynomial $p\left(z\right)=p(z_1,\dots ,z_d)$ with a determinantal representation$p\left(z\right)=p\left(0\right)\det (I_n-K({\oplus }_{j=1}^d{z}_j{I}_{n_j}\left)\right)$ 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 $p\left(z\right)$ to a polynomial in d-tuples of matrices of arbitrary size given by$p(U_1,\dots ,U_d)=p\left(0\right)\det (I_n\otimes I_m-(K\otimes I_m\left)\right({\oplus }_{j=1}^d{I}_{n_j}\otimes U_j\left)\right),$ where $U_1,\dots ,U_d\in C^{m\times m}$, $m\in N$, does not have roots on the noncommutative d-torus consisting of d-tuples $(U_1,\dots ,U_d)$ of unitary matrices of arbitrary size.