Reducibility of rxm+pe f(x) for Large Primes p
Abstract
We study the reducibility over ℚ of Fp,e(x)=rxm+pef(x), where r∈ℤ∖{0}, f∈ℤ[x], and 0≤ m<n:=deg f, for primes p above explicit coefficient-dependent thresholds. For arbitrary e≥1, we determine the degrees, endpoint p-adic valuations, reductions modulo p, and heights of all nonconstant proper integral factors. Writing δ=gcd(e,m,n), we show that every such factor has degree tn/δ for some 1≤ t≤δ−1; in particular, δ=1 implies irreducibility. For e=2 and e=3, we obtain effective necessary and sufficient criteria in terms of at most two associated binary quadratic forms and at most one associated binary cubic form, respectively. The quadratic criterion reduces to at most two fixed Pell-type equations with prescribed second coordinate p. In the cubic case, Thue's theorem yields finiteness of the reducible primes whenever the associated form is irreducible over ℚ.
Related
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- Reducibility of linear representations, free ideals, and Kippenhahn's conjecture
- Reducibility and Diagonal Equality Strata for Finite Jacobi Pencils
BibTeX
@misc{zhang2026reducibility,
title = {{Reducibility of $rx^m+p^e f(x)$ for Large Primes $p$}},
author = {Weilin Zhang and Hongjian Li},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29594},
eprint = {2609.29594},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29594},
}
RIS
TY - UNPB
TI - Reducibility of rx^m+p^e f(x) for Large Primes p
AU - Weilin Zhang
AU - Hongjian Li
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.29594
UR - https://arxiv.org/abs/2609.29594
AB - We study the reducibility over ℚ of F_p,e(x)=rx^m+p^ef(x), where r∈ℤ∖{0}, f∈ℤ[x], and 0≤ m<n:=deg f, for primes p above explicit coefficient-dependent thresholds. For arbitrary e≥1, we determine the degrees, endpoint p-adic valuations, reductions modulo p, and heights of all nonconstant proper integral factors. Writing δ=gcd(e,m,n), we show that every such factor has degree tn/δ for some 1≤ t≤δ−1; in particular, δ=1 implies irreducibility. For e=2 and e=3, we obtain effective necessary and sufficient criteria in terms of at most two associated binary quadratic forms and at most one associated binary cubic form, respectively. The quadratic criterion reduces to at most two fixed Pell-type equations with prescribed second coordinate p. In the cubic case, Thue's theorem yields finiteness of the reducible primes whenever the associated form is irreducible over ℚ.
ER -