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Stable quadratic polynomials over ℚ(i)
Abstract
We study iterates of a quadratic f= x2+1/c∈ K[x]. If the number of factors of fn:=f∘ f ∘ ... ∘ f is bounded by a constant independent of n, then f is said to be \emph{eventually stable}. This paper is an extension to ℚ(i) of the paper \cite{evstb}, which considered f over ℚ. The conjecture "if f2 is irreducible, then fn is irreducible for all n" extends to ℚ(i), but due to the lack of a linear ordering on ℚ(i), an auxiliary function is involved in a specific n to check. The elusive case of c≡ 2 bmod 4 (as a ℤ equivalence class) is shown to be "stable" over ℚ(i), offering more evidence for \cite[Conjecture 1]{evstb}. Stability for c≡ 1bmod 2 (as a ℤ[i] equivalence class) is not as fully handled as over ℤ, however.
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BibTeX
@misc{mcdermott2026stable,
title = {{Stable quadratic polynomials over $\mathbbQ(i)$}},
author = {Jermain McDermott},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.25250},
eprint = {2606.25250},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.25250},
}
RIS
TY - UNPB
TI - Stable quadratic polynomials over ℚ(i)
AU - Jermain McDermott
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2606.25250
UR - https://arxiv.org/abs/2606.25250
AB - We study iterates of a quadratic f= x^2+1/c∈ K[x]. If the number of factors of f^n:=f∘ f ∘ ... ∘ f is bounded by a constant independent of n, then f is said to be \emph{eventually stable}. This paper is an extension to ℚ(i) of the paper \cite{evstb}, which considered f over ℚ. The conjecture "if f^2 is irreducible, then f^n is irreducible for all n" extends to ℚ(i), but due to the lack of a linear ordering on ℚ(i), an auxiliary function is involved in a specific n to check. The elusive case of c≡ 2 bmod 4 (as a ℤ equivalence class) is shown to be "stable" over ℚ(i), offering more evidence for \cite[Conjecture 1]{evstb}. Stability for c≡ 1bmod 2 (as a ℤ[i] equivalence class) is not as fully handled as over ℤ, however.
ER -