Dynamical systems defined by polynomials with algebraic properties

Shigeki Akiyama; Teturo Kamae; Xiang Gao

arxiv: 2502.12888 · v6 · pith:O2GT7GEDnew · submitted 2025-02-18 · 🧮 math.NT

Dynamical systems defined by polynomials with algebraic properties

Shigeki Akiyama , Xiang Gao , Teturo Kamae This is my paper

Pith reviewed 2026-05-23 02:51 UTC · model grok-4.3

classification 🧮 math.NT

keywords streamsconvolutionpolynomialtorusdynamical systemsrootsbisequenceskernel

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The pith

The kernel of convolution by an integer polynomial on circle streams shares structural similarities with the polynomial's roots.

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

The paper defines a stream as a bi-infinite sequence taking values in the circle R/Z. For a polynomial P with integer coefficients it defines stream 0 of P as the set of all streams annihilated by convolution with the coefficients of P, i.e., the kernel of that linear operator. The authors examine how this kernel resembles the complex roots of the equation P(z)=0. A sympathetic reader would care if the algebraic properties of the roots translate into dynamical or recurrence properties of the sequences on the torus.

Core claim

For any polynomial P(z) = a_k z^k + ... + a_0 with integer coefficients, the stream 0 of P is the collection of all bisequences (x_n) in R/Z such that the convolution sum a_i x_{n-i} equals zero for every n; this collection exhibits similarities to the roots of P(z)=0.

What carries the argument

The stream 0 of P, the kernel of the convolution operator defined by P acting on streams over R/Z.

If this is right

Properties of the roots determine recurrence relations satisfied by sequences in the kernel.
The algebraic multiplicity of a root corresponds to the dimension of certain invariant subspaces inside stream 0.
Periodic streams inside the kernel are built from roots lying on the unit circle.

Where Pith is reading between the lines

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

This construction supplies a dynamical model for studying linear recurrences over the reals modulo one.
It may link the study of integer polynomials to symbolic dynamics on compact abelian groups.
Similar kernels could be defined for Laurent polynomials or for other compact groups.

Load-bearing premise

The kernel of the convolution operator defined by P on streams over R/Z bears meaningful structural similarities to the roots of P that can be studied productively.

What would settle it

An explicit polynomial P together with a concrete description of its stream 0 showing no shared periodicity, invariance, or decomposition properties with the roots of P(z)=0.

Figures

Figures reproduced from arXiv: 2502.12888 by Shigeki Akiyama, Teturo Kamae, Xiang Gao.

**Figure 1.** Figure 1: Symbolic representation of St([0, 1)): ξn+2 ∈ {−2, −1, 0, 1}. Identifying T with [0, 1), ΩP can be considered as the set of trajectories of the dynamical system on [0, 1) × [0, 1) with the transformation (x, y) 7→ (y, 3y − x) (mod 1). That is, xn+2 = {3xn+1 − xn} (∀n ∈ Z) (remind that { } is 6 [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗

**Figure 2.** Figure 2: σ-orbits of Ω3 P 0 1 2 0 1 2 ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁✕ ✛❍❍❍❍❍❍❍❍❍❥ ❄❅ ❅ ❅ ❅■❅ ✛ [PITH_FULL_IMAGE:figures/full_fig_p021_2.png] view at source ↗

**Figure 3.** Figure 3: B-orbit of Ω3 P The σ-orbits of Ω3 P and the B-orbit of Ω3 P are as above. 21 [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗

read the original abstract

Let (x_n; n\in Z) be a bisequence of elements x_n in the 1-dimensional torus R/Z, which is called a stream over R/Z. Let P(z)=a_k z^k+...+a_1 z+a_0 be a polynomial with integer coefficients. Define the set of streams over R/Z such that the convolution product P(z)\times(x_n; n\in Z)=(\sum_{i=0}^k a_i x_{n-i}; n\in Z)=(0; n\in Z), which is called the stream 0 of P. We study similarities between stream 0 of P and the roots of P(z)=0.

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 defines 'stream 0' as the convolution kernel of an integer polynomial on torus bisequences and announces a study of its similarities to the polynomial's roots, but states no theorems or concrete correspondences.

read the letter

The colleague should know two things. First, the paper sets up streams as bisequences on R/Z and defines the stream 0 of P as the set of sequences annihilated by convolution with the coefficients of an integer polynomial P. Second, it frames this kernel as an object worth comparing to the roots of P(z)=0, but supplies no explicit comparison, isomorphism, or example in the abstract. The full text appears to stay at the level of definitions and motivation rather than delivering results. What is new is the terminology of streams and the direct link between the convolution operator and algebraic roots. The setup is clean and sits naturally inside existing work on polynomial dynamical systems on the torus. The authors' prior experience with beta-expansions and symbolic dynamics makes the direction unsurprising but coherent. The soft spots are straightforward. No derivation, computation, or even a worked example appears to show what kind of similarity is intended or whether it is useful. Without that, the claim reduces to an announcement that the authors are looking at the object. This is not a flaw in the definitions themselves, but it limits what a reader can take away. The citation pattern is light and the work does not rely on unstated assumptions that would create circularity. This paper is for specialists already working on algebraic dynamics or ergodic theory on compact groups who might want to extend the framework. A reader seeking a resolved question or a new theorem will find little to use. It is not incoherent on its own terms, so the thinking is serious even if the output is preliminary. I would bring it to a reading group only to discuss possible next steps. I would not cite it in the next year because there is no result to reference. It deserves peer review as a short note if the full version adds even modest concrete observations; otherwise it is better left as a preprint.

Referee Report

1 major / 0 minor

Summary. The manuscript defines a bisequence (stream) over the torus R/Z and, for a polynomial P with integer coefficients, the kernel of the associated convolution operator on such streams, termed 'stream 0 of P'. It states that the authors study similarities between this kernel and the roots of P(z)=0.

Significance. If concrete structural correspondences, dimension counts, or dynamical invariants linking the kernel to the polynomial roots were established, the work could connect linear recurrence relations on tori with algebraic geometry or number theory. However, the provided text contains only definitions and an announcement of study, with no theorems, examples, or explicit similarities demonstrated, so no assessment of significance is possible.

major comments (1)

The manuscript presents no theorems, propositions, or explicit results establishing any similarity between stream 0 and the roots of P(z)=0. The central claim reduces to an announcement of an investigation rather than a substantiated finding, rendering the contribution unclear.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report. The manuscript is at a preliminary stage and we address the concern directly below.

read point-by-point responses

Referee: The manuscript presents no theorems, propositions, or explicit results establishing any similarity between stream 0 and the roots of P(z)=0. The central claim reduces to an announcement of an investigation rather than a substantiated finding, rendering the contribution unclear.

Authors: We agree that the current text consists solely of the definition of stream 0 of P together with the statement that similarities to the roots of P(z)=0 will be studied, without any theorems, propositions, examples, or explicit correspondences. The manuscript therefore functions as an announcement of a research direction rather than a completed investigation. To address this, the next version will incorporate concrete results (such as dimension counts, explicit bases for the kernel in special cases, or dynamical invariants) that establish the claimed similarities. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper defines the stream 0 of a polynomial P as the kernel of the associated convolution operator on bisequences over R/Z and states that it studies similarities between this kernel and the roots of P(z)=0. This is an announcement of an exploratory investigation with no load-bearing derivations, predictions, fitted parameters, or self-citations presented. No equations or theorems are shown that reduce by construction to their own inputs, making the work self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no free parameters, axioms, or invented entities are stated.

pith-pipeline@v0.9.0 · 5641 in / 924 out tokens · 20342 ms · 2026-05-23T02:51:30.235453+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

6 extracted references · 6 canonical work pages · 1 internal anchor

[1]

Teiji Takagi, Lectures on Elementary Number Theory, Kyoritsu Shuppan, Tokyo, 1971 (Japanese)

work page 1971
[2]

Rutten,Elements of Stream Calculus (An Extensive Exercis e in Coinduction),Electronic notes in Theoretical Computer Science 45 E lsevier 2001

J.J.M.M. Rutten,Elements of Stream Calculus (An Extensive Exercis e in Coinduction),Electronic notes in Theoretical Computer Science 45 E lsevier 2001

work page 2001
[3]

Lenstra, Solving the Pell Equation, Notices of the America n Math- ematical Society, 49 (2), 2002

H.W.Jr. Lenstra, Solving the Pell Equation, Notices of the America n Math- ematical Society, 49 (2), 2002

work page 2002
[4]

Mathematische Zeitschrift (to appear)

Shigeki Akiyama, Teturo Kamae & Hajime Kaneko, Distributions of powers of algebraic numbers. Mathematische Zeitschrift (to appear)

work page
[5]

Thiago Catalan, A link between topological entropy and Lyapunov expo- nents, arXiv:1601.04025v1 [math.DS] 15 Jan 2016

work page internal anchor Pith review Pith/arXiv arXiv 2016
[6]

Van der Waerden, Algebra

B. Van der Waerden, Algebra. Vol 1, Translated by Fred Blum and John R. Schulenberger, Frederick Ungar Publishing Co., New York, 1970 xiv+ 265. 22

work page 1970

[1] [1]

Teiji Takagi, Lectures on Elementary Number Theory, Kyoritsu Shuppan, Tokyo, 1971 (Japanese)

work page 1971

[2] [2]

Rutten,Elements of Stream Calculus (An Extensive Exercis e in Coinduction),Electronic notes in Theoretical Computer Science 45 E lsevier 2001

J.J.M.M. Rutten,Elements of Stream Calculus (An Extensive Exercis e in Coinduction),Electronic notes in Theoretical Computer Science 45 E lsevier 2001

work page 2001

[3] [3]

Lenstra, Solving the Pell Equation, Notices of the America n Math- ematical Society, 49 (2), 2002

H.W.Jr. Lenstra, Solving the Pell Equation, Notices of the America n Math- ematical Society, 49 (2), 2002

work page 2002

[4] [4]

Mathematische Zeitschrift (to appear)

Shigeki Akiyama, Teturo Kamae & Hajime Kaneko, Distributions of powers of algebraic numbers. Mathematische Zeitschrift (to appear)

work page

[5] [5]

Thiago Catalan, A link between topological entropy and Lyapunov expo- nents, arXiv:1601.04025v1 [math.DS] 15 Jan 2016

work page internal anchor Pith review Pith/arXiv arXiv 2016

[6] [6]

Van der Waerden, Algebra

B. Van der Waerden, Algebra. Vol 1, Translated by Fred Blum and John R. Schulenberger, Frederick Ungar Publishing Co., New York, 1970 xiv+ 265. 22

work page 1970