Category: Number Theory

Summary: Comparing how quickly the count of real roots approaches its central-limit behavior for random polynomials with Gaussian, Rademacher, and uniform coefficients.

Classical results show that the number of real roots of random polynomials has logarithmic-scale mean and variance and obeys a central limit theorem after normalization. This experiment asks whether the rate of convergence to that Gaussian limit depends on the distribution of the coefficients, not just on the degree.

The run samples large random polynomials with Gaussian, Rademacher, and uniform coefficients and compares the resulting real-root counts after centering and variance scaling. By using high sample counts across distributions, it turns an asymptotic theorem into a practical finite-size convergence study.

That comparison is interesting because universality results often emphasize the limiting law itself. Here the emphasis is on how quickly different microscopic coefficient rules approach that common large-degree behavior.

Method: Large-sample simulations of random polynomials across coefficient distributions, followed by normalized real-root count comparisons against the Gaussian CLT prediction.

What is measured: Real-root counts, normalized convergence to Gaussian behavior, dependence on coefficient distribution, and finite-degree convergence rates.