= Reciprocal-conjugate root pairing
For a nonzero real-on-the-circle <trigonometric polynomial> of actual order $d$, the <polynomial> $P(z)=z^dp(z)$ satisfies $P(z)=z^{2d}\overline{P(1/\overline z)}$. Its constant term is the nonzero conjugate of its leading <coefficient>. Hence its nonzero <roots of a polynomial> occur in pairs $\zeta,1/\overline\zeta$ with the same <multiplicity of a root>. The <coefficient> reversal and <complex conjugation> preserve the exponents of the paired factors. Fixed <roots of a polynomial> on the <unit circle> require nonnegativity, rather than this symmetry alone, to have even multiplicity.
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