Rouché localization of polynomial roots (${\sup }_{|z-r_j|=\delta }|H(z)|<\delta {\prod }_{\ell \ne j}(|r_j-r_{\ell }|-\delta )$)

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, $|P(z)|\ge \delta {\prod }_{\ell \ne j}(|r_j-r_{\ell }|-\delta )$. 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.