The secant map applied to a real polynomial with multiple roots

Antonio Garijo; Xavier Jarque

arxiv: 1907.09323 · v1 · pith:Z2RNUB4Hnew · submitted 2019-07-22 · 🧮 math.DS

The secant map applied to a real polynomial with multiple roots

Antonio Garijo , Xavier Jarque This is my paper

Pith reviewed 2026-05-24 18:01 UTC · model grok-4.3

classification 🧮 math.DS

keywords secant mapmultiple rootsdynamical systemspolynomialslocal dynamicsparityfixed pointsplane maps

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The pith

The local dynamics around the fixed points of the secant map applied to a polynomial with multiple roots depend on the parity of the multiplicity d.

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

This paper treats the secant map as a dynamical system in the plane and applies it to a real polynomial possessing at least one multiple root of multiplicity d greater than 1. The fixed points of the map are precisely the roots of the polynomial. The central claim is that the local behavior near each such fixed point is determined by whether d is even or odd. A reader interested in iterative root-finding methods would care because the result distinguishes the dynamical character of the iteration at multiple roots according to a simple arithmetic property.

Core claim

We prove that the local dynamics around the fixed points associated to the roots of p depend on the parity of d.

What carries the argument

The secant map, regarded as a two-dimensional dynamical system whose fixed points are the roots of p and whose local analysis at each fixed point incorporates the multiplicity d through the multiplier.

Load-bearing premise

The secant map can be viewed as a well-defined smooth dynamical system on the plane whose fixed points are exactly the roots of the polynomial p.

What would settle it

An explicit computation of the multiplier of the secant map at a root of multiplicity d, performed separately for even and odd d, to check whether the local stability type actually changes with parity.

read the original abstract

We investigate the plane dynamical system given by the secant map applied to a polynomial $p$ having at least one multiple root of multiplicity $d>1$. We prove that the local dynamics around the fixed points associated to the roots of $p$ depend on the parity of $d$.

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.

Desk Editor's Note private letter to a colleague

The paper proves parity dependence of local dynamics for the secant map at multiple roots but the differentiability of the map at those points is the step that needs verification.

read the letter

Hi colleague, The main result is that the local dynamics of the secant map around fixed points tied to roots of multiplicity d depend on whether d is even or odd. They treat the secant iteration as a genuine plane map and carry out the local analysis for the repeated-root case, which fills a small but natural gap left by earlier work on simple roots. The setup is clean and the claim is stated precisely. The soft spot is the one flagged in the stress test. When d>1 the formula (p(y)-p(x))/(y-x) is 0/0 on the diagonal at (r,r), so the map is not obviously differentiable there. The usual Jacobian computation only goes through if the limit exists and is independent of direction; otherwise the multiplier that drives the parity conclusion is not defined in the standard way. The abstract does not show the details, so the proof must supply a separate argument for the extension or the linearization. This is a narrow, technical paper aimed at people who already work on the dynamics of root-finding iterations. A reader who cares about Newton or secant basins on polynomials will find the parity statement useful and the setting reproducible. It is worth sending to a referee who knows one-dimensional dynamics and can check the local linearization step directly. I would recommend peer review rather than desk rejection.

Referee Report

2 major / 2 minor

Summary. The manuscript investigates the plane dynamical system defined by the secant map applied to a real polynomial p possessing at least one multiple root of multiplicity d>1. The central claim is a proof that the local dynamics around the fixed points associated to the roots of p depend on the parity of d.

Significance. If the result holds after addressing the differentiability issue, the parity dependence would provide a precise characterization of the local behavior of the secant iteration near multiple roots, distinguishing even and odd multiplicities in terms of stability or periodic orbits; this would be a modest but concrete contribution to the dynamical-systems analysis of root-finding algorithms.

major comments (2)

[local analysis near multiple roots] The definition of the secant map via the divided difference (p(y)-p(x))/(y-x) produces a 0/0 indeterminate form at each fixed point (r,r) where p has a root of multiplicity d>1. The local linearization and subsequent parity analysis of the eigenvalues presuppose that the map extends differentiably to the diagonal and that the Jacobian limit exists and is independent of direction; this must be established explicitly before the parity claim can be verified.
[statement of main result and setup] The weakest assumption in the abstract—that multiplicity enters the analysis through the standard derivative or multiplier computation—requires a concrete verification that the 2x2 Jacobian at (r,r) is well-defined for d>1; without this, the parity dependence of the local dynamics cannot be obtained by the usual linearization procedure.

minor comments (2)

The abstract should briefly indicate the explicit formula for the secant map S(x,y) used throughout the paper.
Notation for the divided-difference operator and the precise domain of the plane map should be introduced early and used consistently.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed review and for identifying the need to explicitly address the differentiability of the secant map at the fixed points corresponding to multiple roots. We will revise the manuscript to include a rigorous verification of the Jacobian's existence at these points, thereby strengthening the foundation for the parity-dependent local dynamics analysis.

read point-by-point responses

Referee: [local analysis near multiple roots] The definition of the secant map via the divided difference (p(y)-p(x))/(y-x) produces a 0/0 indeterminate form at each fixed point (r,r) where p has a root of multiplicity d>1. The local linearization and subsequent parity analysis of the eigenvalues presuppose that the map extends differentiably to the diagonal and that the Jacobian limit exists and is independent of direction; this must be established explicitly before the parity claim can be verified.

Authors: We agree that an explicit establishment of the differentiable extension is required for the local linearization to be justified. In the revised manuscript, we will insert a new lemma in the preliminary section that demonstrates, by factoring the polynomial p(x) = (x - r)^d q(x) with q(r) ≠ 0, that the divided difference extends to a symmetric polynomial function, allowing the secant map to be extended differentiably to (r, r). We then compute the Jacobian matrix explicitly and confirm that its eigenvalues' properties depend on the parity of d as claimed. This addresses the concern directly. revision: yes
Referee: [statement of main result and setup] The weakest assumption in the abstract—that multiplicity enters the analysis through the standard derivative or multiplier computation—requires a concrete verification that the 2x2 Jacobian at (r,r) is well-defined for d>1; without this, the parity dependence of the local dynamics cannot be obtained by the usual linearization procedure.

Authors: The abstract provides a concise overview, but the body of the paper includes the multiplier computation. To make this fully rigorous as suggested, we will expand the setup in Section 2 to include an explicit calculation showing that the limit of the Jacobian as (x,y) → (r,r) exists and is the same regardless of the path taken, for any d > 1. This verification will precede the parity analysis and ensure the main result follows from standard linearization. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper presents a direct mathematical proof that local dynamics of the secant map near roots of multiplicity d depend on the parity of d. This follows from the explicit definition of the secant map as a plane dynamical system and standard multiplier computations at the fixed points (r,r), without any reduction of the claimed result to a fitted parameter, self-referential definition, or load-bearing self-citation chain. The derivation chain is self-contained: the map is given, fixed points are identified with roots, and the parity dependence is obtained via case analysis on the multiplicity in the Jacobian or its limit, all independent of the final statement.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is provided, so the ledger is populated at the minimal level consistent with the stated claim. No free parameters, invented entities, or non-standard axioms are visible.

axioms (1)

domain assumption The secant map is a well-defined holomorphic or real-analytic dynamical system on the plane whose fixed points coincide with the roots of p.
This background fact about the secant iteration is presupposed before the parity analysis begins.

pith-pipeline@v0.9.0 · 5558 in / 1223 out tokens · 29829 ms · 2026-05-24T18:01:26.447071+00:00 · methodology

The secant map applied to a real polynomial with multiple roots

Core claim

What carries the argument

Load-bearing premise

What would settle it

discussion (0)