Category: Number Theory

Summary: Testing whether intermediate AR(1) coefficient correlation creates a finite-size window with unusually strong real-zero clustering near the edges of Kac polynomials.

Random polynomials with independent coefficients have well-studied asymptotic root statistics, but finite-size structure can change when coefficient correlations are introduced. This experiment asks whether AR(1)-correlated Gaussian coefficients create an intermediate-correlation regime in which real zeros cluster especially strongly near |x| close to 1.

The GPU workflow evaluates large ensembles of Kac polynomials and measures both normalized real-root counts and edge-localized zero density. The key question is whether those observables become nonmonotone in the correlation strength rather than interpolating smoothly between independent and nearly deterministic limits.

That would expose a finite-size crossover not captured by standard asymptotic theory. The project is therefore aimed at mapping a correlation-driven edge-clustering window rather than only reproducing known large-degree averages.

Method: GPU batched evaluation of Kac polynomials with AR(1)-correlated Gaussian coefficients, sweeping correlation strength at large degree.

What is measured: Normalized real-root count, density of real zeros near |x| approximately 1, correlation-dependent crossover structure, and finite-size enhancement window.