Finite-Degree Probabilistic Zero Certificates for Random Polynomials

Classical zero-localization theorems give deterministic certificates that all zeros of a polynomial lie in a prescribed disk, annulus, or related region. When the coefficients are random, each such deterministic certificate becomes a random variable on coefficient space. This paper develops a finite-degree certificate method for random polynomial localization and extends the author's earlier joint work with Mir \cite{SheikhMir2024}. The main result concerns Gaussian polynomials with random leading coefficient: the Cauchy ratios are marginally standard Cauchy, yet they are dependent through their common denominator. We derive the exact dependence-aware certificate integral and prove that its inverse confidence radius has order \(\sqrt{\log n}\), while a fictitious independent-Cauchy model has order \(n\). We also obtain monic coefficient-law certificates, sub-Weibull confidence radii, annular certificates via reversal, Rouch\'e--Chernoff certificates, and an optimized scaled Cauchy envelope. A reproducible Monte Carlo study for monic Gaussian polynomials compares the classical Cauchy radius, the optimized Cauchy envelope, the annular certificate, and the Rouch\'e radius. Across \(5000\) samples for each of \(n=20,50,100\), the Rouch\'e certificate gives the sharpest outer radii, the optimized Cauchy envelope substantially improves the classical Cauchy radius, and all tested certificates satisfy their deterministic containment inequalities numerically.