Exact Order of the Ho–Nguyen Polynomial System for Overfitted Gaussian Mixtures

Abstract

This paper determines the exact order of the polynomial system introduced by Ho and Nguyen in the study of singularities and parameter convergence rates for overfitted Gaussian mixtures. The result resolves the general order predicted from the previously known two- and three-component cases. The upper bound is proved by converting coefficient cancellation in the polynomial system into vanishing moments of a signed finite combination of Gaussian densities. A sharp bound on the number of sign changes of such Gaussian combinations is then combined with a sign-interpolating polynomial to rule out any nontrivial solution at the critical order. The argument applies to arbitrary positive mixture weights and arbitrary real location and scale perturbation parameters. Sharpness is obtained from positive Gaussian quadrature. For every number of excess components, a Gauss–Hermite construction gives a nontrivial positive-weight configuration attaining cancellation through the preceding order. Thus the upper and lower bounds coincide for every component count covered by the Ho–Nguyen system. The proof is self-contained at the level needed for the main result and includes a treatment of repeated Gaussian variances in the sign-change argument. Exact symbolic computations provide independent checks of the quadrature identities, low-dimensional cases, and coefficient formulas, but the general theorem is analytic and does not depend on finite computational verification. The result concerns the Ho–Nguyen polynomial system associated with overfitted Gaussian location-scale mixtures. It is distinct from the author's earlier work on skew-normal mixture paths, although both use positive Gauss–Hermite quadrature as a sharp lower-bound mechanism. No claim of priority over all unpublished or undiscovered work is made. Research methodology and AI assistance:This work was developed using the CARMA-Math research workflow, a cumulative AI-assisted mathematical research methodology using persistent research archives, literature and prior-art investigation, iterative proof exploration, and verification procedures. Generative AI (ChatGPT) was used extensively for mathematical exploration, proof development, computational reasoning, literature research, and manuscript preparation.