Category: Number Theory

Summary: Testing whether moderate sign persistence in random Kac polynomials concentrates more real roots near x = +/-1 than either independent or highly persistent signs.

Random Kac polynomials are a classic setting for studying how coefficient statistics shape root distributions. This experiment asks whether introducing Markov sign persistence creates a useful middle regime in which real roots cluster more strongly near the edge points x = +/-1 without collapsing the broader nonreal root cloud toward the real axis.

The simulation draws random polynomial coefficients with persistent sign structure and compares root statistics across persistence strengths. The target is a crossover rather than a monotone threshold: weak persistence may do too little, while very strong persistence may over-structure the coefficients and reduce the effect.

That makes the experiment about nontrivial balance in coefficient correlations. It probes whether short-memory sign dependence can produce edge-root enhancement that is not visible in the standard independent-sign ensemble.

Method: Repeated random-polynomial root solves comparing edge-root concentration and complex-root geometry across Markov sign-persistence strengths.

What is measured: Real-root mass near x = +/-1, nonreal-root cloud geometry, crossover strength, and condition-level comparisons.