= Rouché localization of polynomial roots
{c}
{title2=$\sup_{|z-r_j|=\delta}|H(z)|<\delta\prod_{\ell\ne j}(|r_j-r_\ell|-\delta)$}
Let $P(z)=\prod_\ell(z-r_\ell)$ have distinct roots. Choose a disc around $r_j$ with radius $\delta$ smaller than every separation from $r_j$. On its boundary, $|P(z)|\geq\delta\prod_{\ell\ne j}(|r_j-r_\ell|-\delta)$. If a <holomorphic> perturbation $H$ satisfies the displayed inequality, <Rouché's theorem> shows that $P+H$ has exactly one root in that disc, counted with multiplicity. Discs confined to specified half-planes or quadrants give direct geometric root-location proofs without needing a global root-continuation argument.
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