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arxiv: 2510.01877 · v3 · submitted 2025-10-02 · 🧮 math.DS

On intertwined polynomials

Fedor Pakovich This is my paper

Pith reviewed 2026-05-18 11:01 UTC · model grok-4.3

classification 🧮 math.DS MSC 37F10
keywords intertwined polynomialsperiodic curvesJulia setsproduct endomorphismsFavre-Gauthier conjecturecomplex dynamics
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0 comments X

The pith

The set Inter(A) of polynomials intertwined with a given A has the structure described by the Favre-Gauthier conjecture.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper proves the Favre-Gauthier conjecture on the structure of Inter(A) for a polynomial A of degree at least two. Inter(A) collects every polynomial B such that some iterate of B is intertwined with some iterate of A. Two polynomials are intertwined when the product endomorphism (A1, A2) on CP1 times CP1 admits an irreducible periodic curve that is neither vertical nor horizontal. A sympathetic reader would care because the result classifies which polynomial dynamical systems are linked through shared periodic behavior on the product space. The paper also bounds the periods of such curves by the sizes of the symmetry groups of the Julia sets of A1 and A2.

Core claim

We prove a conjecture of Favre and Gauthier describing the structure of Inter(A). We also obtain a bound on the possible periods of periodic curves for endomorphisms (A1, A2) in terms of the sizes of the symmetry groups of the Julia sets of A1 and A2.

What carries the argument

The product endomorphism (A1, A2) on CP1 x CP1 together with an irreducible periodic curve that is neither vertical nor horizontal, which defines when two polynomials are intertwined.

If this is right

  • Inter(A) has a specific structure as stated in the Favre-Gauthier conjecture.
  • Periods of periodic curves for product endomorphisms are bounded by the orders of the symmetry groups of the Julia sets.
  • Pairs of polynomials can be classified according to whether iterates satisfy the intertwining condition.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The classification may extend to rational maps or maps on higher-dimensional spaces.
  • Explicit computation of symmetry groups for quadratic polynomials such as z squared plus c could test the period bound directly.
  • The result may connect to questions of rigidity or common features in complex dynamical systems.

Load-bearing premise

The definition of intertwined polynomials assumes that the product endomorphism admits an irreducible periodic curve that is neither vertical nor horizontal.

What would settle it

A concrete pair of polynomials A and B where Inter(A) deviates from the structure given by the Favre-Gauthier conjecture, or a periodic curve whose period exceeds the bound set by the symmetry group sizes of the Julia sets.

read the original abstract

Let $A_1$ and $A_2$ be polynomials of degree at least two over $\mathbb C$. We say that $A_1$ and $A_2$ are intertwined if the endomorphism $(A_1, A_2)$ of $\mathbb C\mathbb P^1 \times \mathbb C\mathbb P^1$ given by $(z_1, z_2) \mapsto (A_1(z_1), A_2(z_2))$ admits an irreducible periodic curve that is neither a vertical nor a horizontal line. We denote by $\mathrm{Inter}(A)$ the set of all polynomials $B$ such that some iterate of $B$ is intertwined with some iterate of $A$. In this paper, we prove a conjecture of Favre and Gauthier describing the structure of $\mathrm{Inter}(A)$. We also obtain a bound on the possible periods of periodic curves for endomorphisms $(A_1, A_2)$ in terms of the sizes of the symmetry groups of the Julia sets of $A_1$ and $A_2$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper defines two polynomials A1 and A2 of degree at least 2 to be intertwined if the product endomorphism (A1, A2) on CP1 × CP1 admits an irreducible periodic curve that is neither vertical nor horizontal. Inter(A) collects all polynomials B such that some iterate of B is intertwined with some iterate of A. The manuscript proves the Favre-Gauthier conjecture giving the structure of Inter(A) and derives a bound on the periods of such periodic curves expressed in terms of the orders of the symmetry groups of the Julia sets of A1 and A2.

Significance. If the proof is correct, the result is significant for complex dynamics: it resolves a conjecture on the classification of intertwined polynomials and supplies an explicit period bound that quantifies the possible dynamics on product spaces. The derivation relies on standard tools of the field (periodic curves, Julia-set symmetries) and supplies both a structural description and a quantitative bound, which together strengthen the contribution.

minor comments (3)
  1. §1, paragraph after Definition 1.1: the phrase 'some iterate of B is intertwined with some iterate of A' would be clearer if the authors explicitly state whether the iterates are taken independently or must share the same period; a short clarifying sentence would remove potential ambiguity for readers.
  2. Theorem 1.3 (the period bound): the statement refers to 'the sizes of the symmetry groups' without recalling the precise notation used for these groups earlier in the text; adding a parenthetical reference to the relevant definition would improve readability.
  3. The proof of the main classification result (presumably in §3 or §4) invokes several standard lemmas from complex dynamics; a brief sentence indicating which of these lemmas are applied verbatim versus which receive minor adaptations would help readers trace the argument.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the positive assessment. The referee's summary correctly identifies the main results: the proof of the Favre-Gauthier conjecture on the structure of Inter(A) and the period bound in terms of symmetry groups of Julia sets. We are pleased that the work is viewed as significant for complex dynamics. As the report recommends minor revision but lists no major comments, we interpret this as an indication that the core arguments are sound. We will make any necessary minor adjustments in the revised version.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained against external conjecture

full rationale

The paper introduces the definition of intertwined polynomials via the existence of an irreducible non-vertical/non-horizontal periodic curve for the product endomorphism (A1, A2) on CP1 x CP1, then defines Inter(A) as the set of B such that some iterate is intertwined with some iterate of A. It proceeds to prove the Favre-Gauthier conjecture on the structure of Inter(A) using standard techniques from complex dynamics. No load-bearing step reduces by construction to a fitted input, self-definition, or self-citation chain; the central result resolves an external conjecture without renaming known results or smuggling ansatze. The derivation remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard background from complex analysis and algebra without introducing new free parameters or postulated entities; the central claim is a proof rather than a derivation from fitted quantities.

axioms (2)
  • standard math Polynomials of degree at least two over the complex numbers define holomorphic endomorphisms of the Riemann sphere CP1.
    Standard setup invoked in the definition of the product endomorphism.
  • domain assumption Periodic curves of the product map can be classified as irreducible and non-vertical/non-horizontal.
    Directly used to define the notion of intertwined polynomials in the abstract.

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Reference graph

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