Irreducible polynomials with restricted digits in base $b(T)$

Abstract

We study monic irreducible polynomials over $\mathbb{F}_q[T]$ whose non-leading digits, with respect to an arbitrary polynomial base $b(T)$, avoid a prescribed set of forbidden digits. Identifying the digit set with the ring $D=\mathbb{F}_q[T]/(b)$, we obtain an asymptotic formula for the number of such irreducible polynomials under conditions given in terms of Fourier parameters of the allowed digit set. The main term contains a singular series measuring the relative density of units among the allowed digits, while the error term is controlled by both pointwise and averaged Fourier estimates. As a consequence, we obtain a general criterion depending only on the cardinality of the forbidden set, as well as stronger results for structured restrictions, including examples in which the forbidden set contains a positive proportion of all digits. In particular, we treat additive cosets, restrictions compatible with the Chinese remainder decomposition of $D$, and coefficient-wise restrictions. The proof adapts the function field circle method for restricted coefficients to arbitrary polynomial bases.