Category: Number Theory

Summary: Probing how edge-root statistics and real-root structure change across a correlation crossover in Kac-style random polynomials with autoregressive coefficients.

Classical Kac polynomials have well-studied root statistics when coefficients are independent, but correlated coefficients can shift where zeros accumulate and how strongly edge regions are populated. This experiment asks whether there is a crossover where edge-root enhancement becomes more pronounced than the interior behavior.

The script generates autoregressive coefficient sequences, computes root summaries, and compares edge and real-axis statistics through low, middle, and high correlation regimes. The main outputs track whether the middle regime suppresses interior structure while amplifying edge effects.

That is scientifically interesting because it moves beyond the standard independent-coefficient setting. Correlated randomness can create qualitatively different spectral geometry, and the experiment is aimed at locating that change directly.

Method: Random-polynomial simulations using autoregressive coefficient generation, followed by root-summary statistics across correlation settings.

What is measured: Edge-window gain, real-window gain, inner suppression, edge-root statistics at low, mid, and high correlation, and positive-fraction support for the crossover.