PreprintCC BY 4.0arXiv
Counting irreducible skew polynomials x2n+axn+1
Abstract
We give a counting formula for the coefficients a∈Fph for which x2n+axn+1 is irreducible in Fph[x;σ], for arbitrary positive integers n and h, where σ(a)=ap and p is an arbitrary prime power. The proof reduces the problem to quadratic reduced norms, establishes a uniformity statement for every Frobenius generator, and counts the remaining coefficients by Möbius inversion.
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BibTeX
@misc{hayata2026counting,
title = {{Counting irreducible skew polynomials $x^2n+ax^n+1$}},
author = {Tomoya Hayata and Takahiro Tsushima},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.04767},
eprint = {2611.04767},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.04767},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Counting irreducible skew polynomials x^2n+ax^n+1
AU - Tomoya Hayata
AU - Takahiro Tsushima
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.04767
UR - https://arxiv.org/abs/2611.04767
AB - We give a counting formula for the coefficients a∈F_p^h for which x^2n+ax^n+1 is irreducible in F_p^h[x;σ], for arbitrary positive integers n and h, where σ(a)=a^p and p is an arbitrary prime power. The proof reduces the problem to quadratic reduced norms, establishes a uniformity statement for every Frobenius generator, and counts the remaining coefficients by Möbius inversion.
ER -