Minimal Realizations and Determinantal Representations in the Indefinite Setting

For a signature matrix J, we show that a rational matrix function M(z) that is strictly J-contractive on the unit circle \({{\mathbb {T}}}\), has a strict \({\tilde{J}}\oplus J\)-contractive realization \(\begin{bmatrix} A &{} B \\ C &{} D \end{bmatrix}\) for an appropriate signature matrix \({\tilde{J}}\); that is, \( M(z) = D +zC (I -zA)^{-1} B \). As an application, we use this result to show that a two variable polynomial \(p(z_1,z_2)\) of degree \((n_1,n_2)\), \(n_2=1\), without roots on \(\{ (0,0) \} \cup ({{\mathbb {T}}} \times \{ 0 \} ) \cup {{\mathbb {T}}}^2\) allows a determinantal representation

$$\begin{aligned} p(z_1, z_2) = p(0,0) \det (I_{n_1+1} - K Z), \ \ Z = z_1 I_{n_1} \oplus z_2 I_{n_2} , \end{aligned}$$(1)

where K is a strict \({\tilde{J}}\oplus J\)-contraction. This provides first evidence of a new conjecture that a two variable polynomial \(p(z_1,z_2)\) of degree \((n_1,n_2)\) has a determinantal representation (1) with K a strict \({\tilde{J}}\oplus J\)-contraction if and only if \(p(z_1,z_2)\) has no roots in \(\{ (0,0) \} \cup {{\mathbb {T}}}^2\).