Universality for Low-Degree Factors of Random Polynomials over Finite Fields.

The article investigates the universality phenomenon for the counts of low-degree irreducible factors of random polynomials over finite fields \(\mathbb{F}_q\) with independent but nonuniform coefficient distributions. It establishes that, under mild assumptions—particularly when \(q = p\) is prime and \(p \leq \exp(n^{1/13})\) where \(n\) is the polynomial degree—the joint distributions of the number of irreducible factors (with or without multiplicity) closely approximate those of uniformly random monic polynomials, measured in total variation distance. The proofs leverage Fourier-analytic techniques and recent advances in the study of the \(ax+b\) process, extending equidistribution results to multiple roots and Hasse derivatives to handle multiplicities. The work also connects these findings to known universality results for random matrices over finite fields and conjectures that similar universal behavior extends to high-degree irreducible factors, supported by numerical simulations. Quantitative bounds and moment-matching arguments underpin the main results, which provide explicit error estimates and apply to various regimes of the finite field size relative to the polynomial degree.