Reducible fibers of polynomials over integers: the Hilbert-Siegel problem over ℚ and its applications
Abstract
Given a polynomial f∈ ℚ[X], the Hilbert--Siegel problem asks for a description of the set of all values t0∈ ℤ, possibly up to a finite set, such that f(X)−t0 is reducible. Reducibility occurs trivially for t0 in the value set of some non-linear left-composition factor of f. The problem therefore reduces to determining the polynomials having infinitely many further reducible values. The biggest step towards a solution so far was taken in joint work of the author with Behajaina and Neftin. We strengthen some of the methods obtained in that work to present the complete solution of the Hilbert--Siegel problem over ℚ. We give applications to the problem of newly reducible iterates of polynomials, and to an unlikely intersection problem on simultaneous reducible fibers of two polynomial maps over ℚ, generalizing a result by Bilu and Tichy.
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BibTeX
@misc{konig2026reducible,
title = {{Reducible fibers of polynomials over integers: the Hilbert-Siegel problem over $\mathbbQ$ and its applications}},
author = {Joachim König},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02434},
eprint = {2611.02434},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.02434},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Reducible fibers of polynomials over integers: the Hilbert-Siegel problem over ℚ and its applications
AU - Joachim König
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.02434
UR - https://arxiv.org/abs/2611.02434
AB - Given a polynomial f∈ ℚ[X], the Hilbert--Siegel problem asks for a description of the set of all values t_0∈ ℤ, possibly up to a finite set, such that f(X)−t_0 is reducible. Reducibility occurs trivially for t_0 in the value set of some non-linear left-composition factor of f. The problem therefore reduces to determining the polynomials having infinitely many further reducible values. The biggest step towards a solution so far was taken in joint work of the author with Behajaina and Neftin. We strengthen some of the methods obtained in that work to present the complete solution of the Hilbert--Siegel problem over ℚ. We give applications to the problem of newly reducible iterates of polynomials, and to an unlikely intersection problem on simultaneous reducible fibers of two polynomial maps over ℚ, generalizing a result by Bilu and Tichy.
ER -