Radial crossing test for polynomial root counts ($\frac{d}{dt}\log |z(t)|=\mathrm{Re}\frac{1}{z{p}^{\prime }(z)}$)

If $p(z(t))=t$ and $p^{\prime }(z)\ne 0$, implicit differentiation gives $z^{\prime }(t)=1/p^{\prime }(z)$. 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.