Irreducible polynomials with restricted digits in base b(T)
Abstract
We study monic irreducible polynomials over 𝔽q[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=𝔽q[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.
Related
- Reduction of integer tiles via CRT and base-p digits
- An irreducible decomposition of the Weil representation restricted to open compact subgroups
- Counting irreducible skew polynomials x2n+axn+1
- Monogenity of the irreducible factors of (k,t)-Fibonacci and (k,t)-Lucas polynomials
- Superpolynomially Large Support in Every Irreducible Factor of Lacunary Polynomials
BibTeX
@misc{arevalo2026irreducible,
title = {{Irreducible polynomials with restricted digits in base $b(T)$}},
author = {Juan Arévalo and Matilde Lalín},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25367},
eprint = {2609.25367},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25367},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Irreducible polynomials with restricted digits in base b(T)
AU - Juan Arévalo
AU - Matilde Lalín
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.25367
UR - https://arxiv.org/abs/2609.25367
AB - We study monic irreducible polynomials over 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=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.
ER -