A Triplet-Based Parameterization for the Local Asymptotic Characterization of Polynomial Roots

This paper introduces a compact three-parameter framework for characterizing the local geometry of polynomial roots. For each root, the framework records its position, its algebraic multiplicity, and a newly defined quantity called the characteristic deflection distance. This third parameter acts as a natural geometric scale: it measures how sharply or gradually the polynomial departs from zero in the immediate vicinity of the root, and it encodes the collective influence of all other roots through their distances from the one being analyzed. The characteristic deflection distance generalizes the classical condition number of a simple root to roots of arbitrary multiplicity, and it allows direct geometric comparison across roots of different degrees. A key finding is that multiplicity alone does not determine geometric dominance — a lower-multiplicity root can have a larger spatial footprint than a higher-multiplicity one, depending on the global root configuration. Symbolic and numerical Python implementations are provided, along with a worked example. The framework extends naturally to polynomials over the complex numbers.