= Even multiplicity of unit-circle roots of a nonnegative trigonometric polynomial
If $P=z^dp$ and $p$ is nonnegative on the <unit circle>, a <root of a polynomial> $\zeta=e^{i\theta_0}$ there has even <multiplicity of a root>. Indeed $p(e^{i\theta})$ is a <real analytic> nonnegative function, whose first nonzero term in its <Taylor series> at a zero must have even order. The map $\theta\mapsto e^{i\theta}-\zeta$ has a simple zero, and multiplication by $e^{-id\theta}$ is nonvanishing, so the order equals the <polynomial> multiplicity.
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