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We show that the counts of low-degree irreducible factors of a random polynomial   $f$   over   $\mathbb {F}_q$   with independent but nonuniform coefficients behave like that of a uniform random polynomial, exhibiting a form of universality for random polynomials over finite fields. Our strongest results require various assumptions on the parameters, but we are able to obtain results requiring only   $q=p$   a prime with   $p\leq \exp ({n^{1/13}})$   where   $n$   is the degree of the polynomial. Our proofs use Fourier analysis and rely on tools recently applied by Breuillard and Varjú to study the   $ax+b$   process, which show equidistribution for   $f(\alpha )$   at a single point. We extend this to handle multiple roots and the Hasse derivatives of   $f$  , which allow us to study the irreducible factors with multiplicity.

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