Unified Proof of Sendov's Conjecture via the Potential Stability Paradigm: Topological Rigidity and Field Monotonicity

Sendov’s Conjecture (1962) asserts that if all zeros {𝑧1,…, 𝑧𝑛} of a polynomial 𝑃(𝑧) of degree n ≥2 lie within the closed unit disk 𝐷, then each zero 𝑧𝑘 is at a distance no greater than 1 from at least one critical point 𝜉 of 𝑃(𝑧). While recent advancements have confirmed the conjecture for sufficiently large 𝑛, a unified proof for intermediate degrees has remained an open challenge due to the complex discrete interactions of root configurations. This paper provides a complete resolution of Sendov’s Conjecture for all 𝑛 ≥ 2 by reformulating the problem within the framework of Logarithmic Potential Dynamics. We treat the critical points as equilibrium positions in a complex force field generated by the zeros of the polynomial. By isolating a fixed root 𝑧1 = 𝑎 ∈ [0,1], the total force 𝐹(𝑧) = 𝑃′(𝑧)/𝑃(𝑧) is decomposed into a Local Attraction Force and a Collective Cloud Repulsion Force. By employing a Laurent series expansion of the cloud force, we derive a universal upper bound for the remainder 𝑅𝑛(ξ) based on the majorization of the spectral moments (power sums) of the roots constrained within 𝐷. We prove that the repulsion magnitude is strictly dominated by the geometric decay of the local force at the boundary |𝑧1 − ξ| = 1. Furthermore, by analyzing the Hessian of the potential, we establish the strict monotonicity of the radial force field in the region |𝑧1 − ξ| > 1, thereby precluding the existence of equilibrium points beyond the unit radius of any given zero. This analytical framework bridges the gap between asymptotic analysis and discrete root geometry, confirming the conjecture for the general case.