Determinantal representation of trigonometric polynomial curves via Sylvester method

Abstract

For any trigonometric polynomial $\phi(\theta)$, we give a constructive algorithm by Sylvester elimination which produces matrices $C_1, C_2, C_3$ such that $\det(C_1+ \Re(\phi(\theta)) C_2+ \Im(\phi(\theta)) C_3)=0.$ For a typical trigonometric polynomial, we assert that $C_1$ is positive definite, and thus the typical polynomial curve admits