Sendov conjecture proof

Title: General Proof of Sendov's Conjecture Description: This paper provides a comprehensive proof of Sendov's conjecture for all degrees of monic polynomials with zeros in the unit disk. The conjecture states that for every zero of such a polynomial, there exists a critical point (zero of the derivative) within distance one. The proof combines classical and modern tools, including Newton's identities, Jacobian formulas, Vandermonde matrices, density lemmas, blow-up techniques, and detailed $\varepsilon$–$\delta$ control of zeros. The case $n=9$ is treated explicitly, with all intermediate lemmas and combinatorial structures detailed. The argument is then extended to all degrees via descending induction, relying on known results for lower-degree polynomials. A supplementary section provides rigorous treatment of small perturbations of zeros, ensuring the robustness of critical point behavior. This work is intended for researchers in complex analysis, algebra, and polynomial dynamics, offering a detailed and self-contained argument for Sendov's conjecture. Author: Zakarya Benregreg Contact: [e-mail] 00213556142305