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Preprint
Indefinite determinantal representations versus nonsingularities on the noncommutative d-torus
08 Nov 2024
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.
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Details
- Title
- Indefinite determinantal representations versus nonsingularities on the noncommutative d-torus
- Creators
- Gilbert J GroenewaldSanne ter HorstHugo J Woerdeman
- Resource Type
- Preprint
- Language
- English
- Academic Unit
- Mathematics
- Other Identifier
- 991021959779804721