Determinantal representation of trigonometric polynomial curves via Sylvester method (Q380313)

Let, for an \(n \times n\) complex matrix \(A\), \(\; F_A(t,x,y) := \det(t \, I_n+x \, \mathfrak{R}(A)+y \, \mathfrak{I}(A))\) denote the real ternary form associated with the matrix \(A\), and let \[ \phi(\theta) = \sum_{k=-n}^n c_k \exp (ik\theta) \] be a trigonometric polynomial. Using Sylvester matrices, these authors present an algorithm to construct \(2n \times 2n\) matrices \(C_1\), \(C_2\), \(C_3\) such that, for a given trigonometric polynomial \(\phi(\theta),\) \[ \det (C_1 + \mathfrak{R}(\phi(\theta)) C_2 + \mathfrak{I} (\phi(\theta)) C_3) = 0. \] If \(\phi\) has the special form \(\phi(\theta) = \exp(in\theta) + a \exp(-im\theta)\), \(0 \leq \theta \leq 2\pi\), \(0 < m < n\), \(0 < a < 1\), then the matrices \(C_1\), \(C_2\), \(C_3\) are Hermitian, \(C_1\) is positive definite, and the relation \[ F_{C_0}(t,x,y) \det(C_1) = \det(tC_1+xC_2+yC_3), \] holds, where \(C_0 = C_1^{-1/2} (C_2+iC_3) C_1^{-1/2}.\)