arxiv: 2605.04417 · v1 · submitted 2026-05-06 · 🧮 math.NT · math.DS

Recognition: unknown

Fixed-point lifting and ghost periodic points for Chebyshev polynomials modulo odd prime powers

Aram Tangboonduangjit, Chatchawan Panraksa

Pith reviewed 2026-05-08 17:33 UTC · model grok-4.3

classification 🧮 math.NT math.DS MSC 11T5537P35

keywords Chebyshev polynomialsfixed pointsperiodic pointsprime power modulip-adic liftingghost periodic pointsChebyshev order

0 comments

The pith

Chebyshev polynomials have explicit fixed-point counts modulo odd prime powers that lift according to the valuation of n squared minus one.

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

This paper counts the fixed points and exact periodic points of the nth Chebyshev polynomial acting on the integers modulo p to the k for odd primes p. It begins with an explicit four-GCD formula for the count over the prime field, separating contributions from split and nonsplit source groups. For higher powers when p is at least 5, the total number of fixed points equals N1 plus d times p to the min of k minus 1 and the p-adic valuation of n squared minus one, where d counts residues with derivative congruent to 1 mod p. The same approach yields closed-form counts for periodic points of all periods via Mobius inversion on the iterates, and describes orbit lifting from modulo p to p squared, which either keeps the period or adds one Hensel lift together with ghost points of period given by the Chebyshev order over e p.

Core claim

Over F_p the fixed-point count N1 equals (gcd(n-1,p-1) + gcd(n+1,p-1) + gcd(n-1,p+1) + gcd(n+1,p+1) - 2 delta)/2 with delta = gcd(n-1,2). For every odd p, N2 equals N1 plus d times (p-1), where d is the number of fixed residues a in F_p with T_n prime congruent to 1 mod p. For p at least 5 and all k at least 1, Nk equals N1 plus d times (p to the min(k-1, v_p(n squared minus 1)) minus 1). A source-order-e point is periodic over F_p exactly when gcd(n,e)=1 with period cord_e(n), and orbitwise lifting modulo p squared either retains the full period or produces one Hensel lift plus ghost periodic points of period cord_{e p}(n).

What carries the argument

The Chebyshev order cord_e(n) equals the minimal r at least 1 such that n to the r is congruent to plus or minus 1 modulo e, which determines the period of source-order-e points when gcd(n,e)=1, together with the splitting of fixed residues into four source groups a = (zeta + zeta inverse)/2 over the algebraic closure of F_p.

If this is right

Mobius inversion applied to the fixed-point counts of the iterates T_{n^j} produces exact-period point counts over Z/p^k Z for all odd p.
For p at least 5 the all-level fixed-point formula directly supplies closed forms for those exact-period counts.
When p does not divide n, orbitwise lifting modulo p squared gives either full period retention or one Hensel lift plus ghost periodic points of period cord_{e p}(n).
For p at least 5, higher lifts of periodic residues are governed by the tower of Chebyshev orders cord_{e p^q}(n).

Where Pith is reading between the lines

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

The explicit formulas allow complete description of the periodic structure of Chebyshev maps over the p-adic integers for p at least 5.
The same source-group and order machinery may extend to counting periodic points for other polynomials arising from endomorphisms of algebraic groups.
Direct computation for small n and p=3 can test where the boundary estimates first fail and whether a modified lifting rule applies.

Load-bearing premise

The p-adic estimates at the boundary points a equal to plus or minus one remain controlled for p at least 5, so that the Chebyshev order and derivative condition determine all lifting behavior without extra degeneracies for odd n at least 2.

What would settle it

Direct enumeration of roots of T_n(x) minus x modulo 5 cubed for n=3, which the formula predicts should equal N1 plus d times (5 to the min(2, v_5(8)) minus 1), compared against the actual count of solutions in Z/125Z.

read the original abstract

Let $p$ be an odd prime, let $n\ge2$, and let the $n$th Chebyshev polynomial $T_n$ act on $\Z/p^k\Z$. We count fixed and exact-periodic points, allowing non-permutation degrees, and organize the finite-field formulas by the two source groups needed for prime-power lifting. Over $\Fp$ we record the four-GCD fixed-point formula \[ N_1=\frac{\gcd(n-1,p-1)+\gcd(n+1,p-1)+\gcd(n-1,p+1)+\gcd(n+1,p+1)-2\delta}{2}, \] where $\delta=\gcd(n-1,2)$. The proof separates split and nonsplit source groups for $a=(\zeta+\zeta^{-1})/2$ and counts degenerate fixed residues branch-wise. For every odd $p$, \[ N_2=N_1+d(p-1). \] Here $d$ denotes the number of fixed residue classes $a\in\Fp$ for which $T_n'(a)\equiv1\pmod p$. For $p\ge5$ and all $k\ge1$, \[ N_k=N_1+d\bigl(p^{\min(k-1,\nup(n^2-1))}-1\bigr). \] This all-level formula does not extend unchanged to $p=3$, where boundary $p$-adic estimates at $a=\pm1$ can fail; the first-lift formula remains valid. For periods, we use the Chebyshev order \[ \cord_e(n)=\min\{r\ge1:n^r\equiv\pm1\pmod e\}. \] A source-order-$e$ point is periodic over $\Fp$ exactly when $\gcd(n,e)=1$, with period $\cord_e(n)$. M\"obius inversion for the iterates $T_{n^j}$ gives exact-period point counts over $\Z/p^k\Z$ for all odd $p$; for $p\ge5$, the all-level fixed-point formula gives closed forms. When $p\nmid n$, orbitwise lifting modulo $p^2$ gives either full period retention or one Hensel lift plus ghost periodic points of period $\cord_{ep}(n)$. For $p\ge5$, higher lifts above a periodic residue are governed by the tower $\cord_{ep^q}(n)$.

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 gives explicit all-level lifting formulas for fixed points of Chebyshev polynomials over Z/p^k Z when p >= 5, plus a workable treatment of periodic points that includes ghost lifts.

read the letter

The core contribution is a set of closed-form counts for fixed points N_k of T_n modulo p^k. Over F_p they start from the four-GCD expression that splits the source groups, then introduce d as the count of residues where the derivative is 1 mod p. For p >= 5 this lifts cleanly to N_k = N_1 + d (p^min(k-1, v_p(n^2-1)) - 1) for all k. The periodic side uses the Chebyshev order cord_e(n) and Möbius inversion on the iterates, with orbitwise rules mod p^2 that either keep the full period or produce ghost points of period cord_ep(n). When p does not divide n the higher lifts follow the tower of orders. These formulas are concrete and avoid free parameters, which is the main practical gain over generic Hensel arguments. The authors are straightforward about the p = 3 limitation at a = ±1, where the all-level formula breaks but the first-lift step still holds. That boundary handling looks like the only real case split. The rest of the argument stays within standard GCD identities, valuations, and derivative conditions, so the derivations appear reproducible from the stated steps. The work is narrow but the explicitness makes the counts immediately usable for anyone computing orbits or testing lifting in arithmetic dynamics. It is not reorganizing the field, yet the formulas are sharper than what is in the usual references for this family. I would send it to referees; the claims are specific enough to verify directly and the gaps are already flagged rather than hidden.

Referee Report

0 major / 3 minor

Summary. The manuscript derives explicit counts for fixed points N_k of the nth Chebyshev polynomial T_n acting on Z/p^k Z, for odd prime p and n >= 2. Over F_p it gives the four-GCD formula N_1 = [gcd(n-1,p-1) + gcd(n+1,p-1) + gcd(n-1,p+1) + gcd(n+1,p+1) - 2 delta]/2 with delta = gcd(n-1,2), obtained by separating split and nonsplit source groups. It then states N_2 = N_1 + d(p-1) where d counts fixed a in F_p with T_n'(a) ≡ 1 mod p, and for p >= 5 the closed form N_k = N_1 + d (p^{min(k-1, v_p(n^2-1))} - 1). Periodic-point counts are obtained via the Chebyshev order cord_e(n) = min{r >= 1 : n^r ≡ ±1 mod e}, Mobius inversion on iterates T_{n^j}, and orbitwise lifting rules that produce ghost periodic points of period cord_{ep}(n) when p does not divide n.

Significance. If the derivations hold, the paper supplies closed-form, parameter-free expressions for fixed-point and exact-period counts in Chebyshev dynamics over odd prime-power rings, together with precise lifting rules from F_p to higher powers. The separation into source groups, the derivative condition defining d, and the explicit treatment of the p = 3 boundary case constitute a structured contribution to arithmetic dynamics. The use of GCD identities and Mobius inversion yields falsifiable, directly computable formulas that can be checked for small p and n.

minor comments (3)

[Abstract] Abstract, N_1 formula: the factor of 1/2 and the precise role of delta = gcd(n-1,2) are stated without an accompanying sentence explaining why the expression is always an integer; a one-line integrality remark would remove any reader uncertainty.
[Periodic points] Periodic-points paragraph: the phrase 'ghost periodic points of period cord_{ep}(n)' is introduced without a parenthetical gloss or forward reference to the precise definition used later in the text; adding a brief inline clarification would improve readability.
[Notation] Notation: the symbol cord_e(n) is defined once but appears in several contexts (source-order, lifting, ghost points); a short table or sentence listing the distinct uses would help readers track the quantity across the lifting statements.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful summary of the manuscript, recognition of its significance in providing closed-form counts and lifting rules for Chebyshev dynamics over odd prime-power rings, and recommendation for minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper derives N_1 via an explicit four-term GCD formula over F_p, defines the auxiliary d as the count of residues satisfying the derivative condition T_n'(a)≡1 mod p, and states the higher-power lifting N_k = N_1 + d(p^min(...) - 1) as a direct consequence of separate p-adic estimates at split/nonsplit sources for p≥5. Periodic counts are obtained by Möbius inversion applied to the independently defined Chebyshev order cord_e(n), with orbitwise lifting rules following from Hensel's lemma. None of these steps reduce a claimed result to a fitted parameter or self-referential definition; the formulas remain expressed in terms of standard arithmetic invariants without load-bearing self-citation chains.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The results rest on standard properties of Chebyshev polynomials, Hensel's lemma for lifting, and Mobius inversion over the poset of divisors; the only novel device is the terminology of ghost periodic points introduced to label certain lifting outcomes.

axioms (2)

domain assumption Chebyshev polynomials satisfy the standard recurrence and trigonometric identities that allow reduction of fixed-point equations to cyclotomic conditions over finite fields.
Invoked when deriving the four-GCD formula and when separating split versus nonsplit source groups.
domain assumption Hensel's lemma applies to lift solutions when the derivative is not congruent to zero, with controlled failure only at a = plus or minus 1 for p = 3.
Used to justify the lifting formulas and the appearance of ghost points.

invented entities (1)

ghost periodic points no independent evidence
purpose: To label the additional periodic points of period cord_{e p}(n) that appear when an orbit lifts to exactly one genuine periodic point plus extras rather than retaining the full period.
Introduced to describe the orbitwise lifting behavior modulo p^2 and higher towers.

pith-pipeline@v0.9.0 · 5771 in / 1691 out tokens · 61512 ms · 2026-05-08T17:33:38.539256+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

22 extracted references · 12 canonical work pages

[1]

A. W. Bluher,Permutation properties of Dickson and Chebyshev polynomials with connections to number theory, Finite Fields Appl.76(2021), Article 101899, doi:10.1016/j.ffa.2021.101899. Preprint arXiv:1707.06877

work page doi:10.1016/j.ffa.2021.101899 2021
[2]

Diarra, D

B. Diarra, D. Sylla,p-adic dynamical systems of Chebyshev polynomials,p-Adic Numbers Ultramet- ric Anal. Appl.6(2014), no. 1, 21–32, doi:10.1134/S2070046614010026; erratum,8(2016), no. 1, 87, doi:10.1134/S2070046616010064. 36 CHATCHA W AN PANRAKSA AND ARAM TANGBOONDUANGJIT

work page doi:10.1134/s2070046614010026 2014
[3]

L. E. Dickson,The analytic representation of substitutions on a power of a prime number of letters with a discussion of the linear group, Ann. of Math.11(1896), 65–120
[4]

S. Fan, L. Liao,Dynamical structures of Chebyshev polynomials onZ 2, J. Number Theory169(2016), 174–182, doi:10.1016/j.jnt.2016.05.014

work page doi:10.1016/j.jnt.2016.05.014 2016
[5]

T. A. Gassert,Chebyshev action on finite fields, Discrete Math.315–316(2014), 83–94, doi:10.1016/j.disc.2013.10.014

work page doi:10.1016/j.disc.2013.10.014 2014
[6]

B. Hutz, T. Patel,Periodic points and tail lengths of split polynomial maps modulo primes, Involve15(2022), no. 2, 185–206, doi:10.2140/involve.2022.15.185. Preprint arXiv:1710.07821

work page doi:10.2140/involve.2022.15.185 2022
[7]

Koblitz,p-adic Numbers,p-adic Analysis, and Zeta-Functions, 2nd ed., Graduate Texts in Mathematics, vol

N. Koblitz,p-adic Numbers,p-adic Analysis, and Zeta-Functions, 2nd ed., Graduate Texts in Mathematics, vol. 58, Springer, 1984

1984
[8]

C. Li, X. Lu, K. Tan, G. Chen,Graph structure of Chebyshev permutation polynomials over ringZ pk, IEEE Trans. Inf. Theory71(2025), no. 2, 1419–1433, doi:10.1109/TIT.2024.3522095

work page doi:10.1109/tit.2024.3522095 2025
[9]

R. Lidl, G. L. Mullen,Cycle structure of Dickson permutation polynomials, Math. J. Okayama Univ.33(1991), 1–11

1991
[10]

R. Lidl, G. L. Mullen, G. Turnwald,Dickson Polynomials, Pitman Monographs, vol. 65, Longman, 1993

1993
[11]

J. C. Mason, D. C. Handscomb,Chebyshev Polynomials, Chapman & Hall/CRC, 2003

2003
[12]

G. L. Mullen, D. Panario (eds.),Handbook of Finite Fields, Discrete Mathematics and Its Applications, CRC Press, Boca Raton, 2013

2013
[13]

Nara,Lifting of cycles in functional graphs, Discrete Math

T. Nara,Lifting of cycles in functional graphs, Discrete Math. Algorithms Appl., online ready (2025), doi:10.1142/S1793830925501708. Preprint arXiv:2509.16234

work page doi:10.1142/s1793830925501708 2025
[14]

Narkiewicz,Polynomial Mappings, Lecture Notes in Mathematics, vol

W. Narkiewicz,Polynomial Mappings, Lecture Notes in Mathematics, vol. 1600, Springer, Berlin, 1995

1995
[15]

Qureshi, D

C. Qureshi, D. Panario,The graph structure of Chebyshev polynomials over finite fields and applications, Des. Codes Cryptogr.87(2019), 393–416, doi:10.1007/s10623-018-0545-7

work page doi:10.1007/s10623-018-0545-7 2019
[16]

T. J. Rivlin,Chebyshev Polynomials: From Approximation Theory to Algebra and Number Theory, 2nd ed., Wiley, 1990

1990
[17]

Rosen, Z

J. Rosen, Z. Scherr, B. Weiss, M. E. Zieve,Chebyshev mappings of finite fields, Amer. Math. Monthly119 (2012), no. 2, 151–155, doi:10.4169/amer.math.monthly.119.02.151

work page doi:10.4169/amer.math.monthly.119.02.151 2012
[18]

J. H. Silverman,The Arithmetic of Dynamical Systems, Graduate Texts in Mathematics, vol. 241, Springer, 2007

2007
[19]

K. Tan, C. Li,The graph structure of a class of permutation maps over ringZ pk, arXiv:2506.20118v1 [math.NT], 2025

work page arXiv 2025
[20]

D. Wan,Ap-adic lifting lemma and its applications to permutation polynomials, inFinite Fields, Coding Theory, and Advances in Communications and Computing(Las Vegas, NV, 1991), Lecture Notes Pure Appl. Math., vol. 141, Dekker, New York, 1993, pp. 209–216

1991
[21]

Q. Wang, J. L. Yucas,Dickson polynomials over finite fields, Finite Fields Appl.18(2012), no. 4, 814–831, doi:10.1016/j.ffa.2012.02.001

work page doi:10.1016/j.ffa.2012.02.001 2012
[22]

Yoshioka,Properties of Chebyshev polynomials modulop k, IEEE Trans

D. Yoshioka,Properties of Chebyshev polynomials modulop k, IEEE Trans. Circuits Syst. II Express Briefs65 (2018), no. 3, 386–390, doi:10.1109/TCSII.2017.2739190. Division of Science, Mahidol University International College, Nakhon Pathom 73170, Thailand Email address:chatchawan.pan@mahidol.ac.th Division of Science, Mahidol University International Colle...

work page doi:10.1109/tcsii.2017.2739190 2018