Category: Number Theory

Summary: Studying whether sign persistence in Kac-polynomial coefficients changes finite-size root crowding near z = +/-1.

Random Kac polynomials usually have roots that cluster near the unit circle, but finite systems can still show special structure near the real-axis edge points z = +/-1. This experiment asks whether that edge behavior changes when the coefficient signs are not independent, but instead follow a two-state Markov process with persistence or anti-persistence.

The script generates many random polynomials with correlated sign patterns and measures how closely roots crowd the edge points before the broader unit-circle pattern dominates. Positive persistence is expected to increase local crowding and shrink the nearest-root gap, while anti-persistence should do the opposite.

That question sits between asymptotic universality and finite-size structure. The experiment focuses on a local geometric effect that may be invisible if one looks only at the global root distribution.

Method: Monte Carlo sampling of Kac polynomials with Markov-sign coefficients, followed by numerical root finding and finite-size edge-clustering analysis.

What is measured: Root crowding near z = +/-1, nearest-root gap to the edge points, dependence on sign persistence, and finite-size crossover behavior.