REVIEW 3 minor 22 references
Stable quadratic polynomials over $\mathbb{Q}(i)$
T0 review · 0 major / 3 minor · reviewed 2026-07-02 · grok-4.3
Pith's one-line read Quadratic polynomials x² + 1/c are eventually stable over Q(i) when c ≡ 2 mod 4.
desk verdict McDermott fills the c ≡ 2 mod 4 stability gap over Q(i) but leaves the odd case incomplete. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
Eventual stability of polynomial iterates, i.e., the property that the number of irreducible factors of the n-fold composition f^n stays bounded by a constant independent of n.
What would settle it
An explicit c ≡ 2 mod 4 together with a concrete large n such that f^n factors into more than the predicted number of irreducibles over Q(i).
Extended reading notes
Core claim
When c belongs to the Z-equivalence class 2 mod 4, the quadratic f = x² + 1/c satisfies eventual stability over Q(i): the number of irreducible factors of f^n remains bounded independently of n. The argument proceeds by showing that if f² is irreducible then all higher iterates remain irreducible, with an auxiliary function supplying the necessary check for a specific n because Q(i) lacks a total order.
Load-bearing premise
That an auxiliary function suffices to verify the irreducibility condition at the needed finite stage even without a linear order on Q(i).
Editorial extensions
If this is right
- If f² is irreducible over Q(i) then every higher iterate f^n remains irreducible.
- The previously unresolved congruence class 2 mod 4 now satisfies the bounded-factor condition.
- The conjecture linking irreducibility of f² to irreducibility of all iterates holds for this class in Q(i).
- Stability results carry over from Q to Q(i) at least for the even-c case.
Reading between the lines
- The same auxiliary-function technique might resolve the remaining odd-c cases in Z[i] with additional computation.
- Analogous stability statements could be tested over other quadratic fields once suitable auxiliary functions are identified.
- The bounded-factor property implies that the dynamical system generated by f has only finitely many distinct prime divisors in its orbit factorizations.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper extends the study of eventual stability for quadratic maps f(x) = x² + 1/c from Q to the Gaussian field Q(i). It proves that the previously elusive case c ≡ 2 mod 4 (as a Z-class) yields eventually stable iterates over Q(i), and discusses the extension of the irreducibility conjecture 'if f² is irreducible then all fⁿ are irreducible' to Q(i), where an auxiliary function is required to check a specific n because Q(i) lacks a linear order. Stability for the c ≡ 1 mod 2 class (now viewed in Z[i]) receives less complete treatment than in the rational case.
Significance. If the central stability result for c ≡ 2 mod 4 holds, the manuscript supplies concrete supporting evidence for Conjecture 1 of the cited prior work by resolving an open congruence class over a new base field. The introduction of an auxiliary function to handle the absence of ordering is a technically natural adaptation, though the incomplete treatment of the odd-c case limits the overall scope.
minor comments (3)
- The abstract states that an auxiliary function is used to check a specific n, but the manuscript should include an explicit definition or reference to its construction in the main text (e.g., near the statement of the extended conjecture).
- The claim that stability for c ≡ 1 mod 2 'is not as fully handled' should be made precise by indicating which subcases remain open and whether they are expected to follow the same argument as over Z.
- Notation for the equivalence classes (Z versus Z[i]) should be clarified at first use to avoid confusion when moving between the rational and Gaussian settings.
Simulated Author's Rebuttal
We thank the referee for their positive assessment and recommendation of minor revision. The report correctly identifies our main contribution (the stability result for c ≡ 2 mod 4) and the technical adaptation needed for the irreducibility conjecture. We address the noted limitation on the odd-c case below.
read point-by-point responses
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Referee: Stability for the c ≡ 1 mod 2 class (now viewed in Z[i]) receives less complete treatment than in the rational case.
Authors: We agree this is a genuine limitation of the present manuscript. The primary goal was to settle the previously open congruence class c ≡ 2 mod 4 over Q(i), which supplies new evidence for Conjecture 1 of the cited work. The odd-c case over Z[i] would require additional tools (e.g., a fuller analysis of primes above 2 and possible ramification in the dynatomic extensions) that go beyond the scope of this short note. We will add a clarifying paragraph in the introduction stating the restricted scope and indicating why a complete treatment is deferred. revision: yes
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Referee: The introduction of an auxiliary function to handle the absence of ordering is a technically natural adaptation, though the incomplete treatment of the odd-c case limits the overall scope.
Authors: We appreciate the referee’s recognition that the auxiliary function is a natural device. It is introduced precisely to replace the ordering argument used over Q when verifying the base case of the irreducibility conjecture for a specific iterate. We will incorporate the referee’s phrasing into a revised introduction to emphasize both the adaptation and the acknowledged limitation on scope. revision: partial
Circularity Check
Minor self-citation to prior conjecture; no reduction of new result to fitted inputs or definitions
full rationale
The paper extends the stability analysis of quadratic iterates from Q to Q(i), proving the c ≡ 2 mod 4 case is stable and thereby supplying additional evidence for Conjecture 1 from the cited prior work. The extension introduces a new number field and an auxiliary function to handle the lack of linear ordering, with the central stability claim presented as an independent verification rather than a quantity defined in terms of the prior result. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citation chains appear in the derivation chain described.
Assumptions & free parameters
Cite this review
Pith. "Pith review of Stable quadratic polynomials over $\mathbb{Q}(i)$." pith.science (2026). https://pith.science/paper/UXGQZZ2Q
@misc{pith2026260625250,
author = {Pith},
title = {Pith review of: Stable quadratic polynomials over $\mathbbQ(i)$},
year = {2026},
howpublished = {\url{https://pith.science/paper/UXGQZZ2Q}},
note = {Machine review of arXiv:2606.25250}
}
abstract
We study iterates of a quadratic $f= x^2+1/c\in K[x]$. If the number of factors of $f^n:=f\circ f \circ ... \circ f$ is bounded by a constant independent of $n$, then $f$ is said to be \emph{eventually stable}. This paper is an extension to $\mathbb{Q}(i)$ of the paper \cite{evstb}, which considered $f$ over $\mathbb{Q}$. The conjecture "if $f^2$ is irreducible, then $f^n$ is irreducible for all $n$" extends to $\mathbb{Q}(i)$, but due to the lack of a linear ordering on $\mathbb{Q}(i)$, an auxiliary function is involved in a specific $n$ to check. The elusive case of $c\equiv 2 \bmod 4$ (as a $\mathbb{Z}$ equivalence class) is shown to be "stable" over $\mathbb{Q}(i)$, offering more evidence for \cite[Conjecture 1]{evstb}. Stability for $c\equiv 1\bmod 2$ (as a $\mathbb{Z}[i]$ equivalence class) is not as fully handled as over $\mathbb{Z}$, however.
Reference graph
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Reviewed July 2, 2026 · model on record in the stance chip above.
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