= Radial crossing test for polynomial root counts
{title2=$\frac d{dt}\log|z(t)|=\operatorname{Re}\frac1{z\,p'(z)}$}
If $p(z(t))=t$ and $p'(z)\ne0$, implicit differentiation gives $z'(t)=1/p'(z)$. At a crossing of the <unit circle>, the displayed radial derivative determines whether the root enters or leaves the <disk> as the real parameter increases. Root counts are constant between such boundary crossings, including multiplicities. At the crossing parameter itself, boundary roots are excluded from an open-disk count. This complements the <winding number> form of the <argument principle> and handles a self-intersecting <image> <contour> without guessing its enclosed regions.
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