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arxiv: 2511.00322 · v2 · submitted 2025-10-31 · 🧮 math.NT · math.DS

Counting the number of n-periodic mathbb{Z}_(p)-and mathbb{F}_(p)[t]-points of a discrete dynamical system with applications from arithmetic statistics, VI

Brian Kintu This is my paper

Pith reviewed 2026-05-18 01:49 UTC · model grok-4.3

classification 🧮 math.NT math.DS
keywords n-periodic pointsp-adic integersfunction fieldsarithmetic dynamicsarithmetic statisticszeta functionsL-functionsirreducible polynomials
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The pith

For the maps z to the power p to the ℓ plus c, the average number of n-periodic points modulo p is unbounded or zero as c tends to infinity.

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

The paper studies the number of distinct n-periodic points for the polynomial maps φ_{d,c}(z) = z^d + c over the p-adic integers and over the field F_p[t]. It establishes that when the exponent d equals p to the power ℓ, this average count over c, taken after reduction modulo p or a prime element π, is either unbounded or zero as c grows large. When d equals (p-1) to the power ℓ and p is at least 5, the average instead stabilizes at one of the values 0, 1, or 2. These counting results are then applied using tools from arithmetic statistics to obtain statements about irreducible polynomials, Artin-Mazur zeta functions, and L-functions attached to the dynamical systems.

Core claim

For any prime p ≥ 3 and fixed ℓ ≥ 1, the average number of distinct n-periodic p-adic integral points of φ_{p^ℓ, c} modulo p Z_p is unbounded or zero as c → ∞; for any prime p ≥ 5 the average number for φ_{(p-1)^ℓ, c} is 1 or 2 or 0 as c → ∞. Analogous statements hold for points over F_p[t] modulo primes π. These limits are obtained by combining density, polynomial counting, and equidistribution results from arithmetic statistics.

What carries the argument

The family of maps φ_{d,c}(z) = z^d + c with d fixed as p^ℓ or (p-1)^ℓ, together with the limit of the average count of n-periodic points after reduction modulo p (or π) as c tends to infinity through integers or polynomials.

If this is right

  • The average number of n-periodic points after reduction is controlled by the arithmetic nature of the exponent relative to the prime p.
  • Equidistribution and density results from arithmetic statistics yield counting theorems for irreducible polynomials in the function-field setting.
  • The periodic-point data produce explicit expressions or limits for Artin-Mazur zeta functions and associated L-functions over global fields.
  • The same averaging technique applies uniformly to both p-adic and function-field cases when the exponent takes the indicated special form.

Where Pith is reading between the lines

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

  • The restriction to these two families of exponents may indicate that the reduction behavior modulo p is governed by the multiplicative order or the p-power structure in the residue field.
  • If similar limits hold for other exponents that share the same residue-field order properties, the results could extend beyond the cases proved here.
  • The connection to torsion-point counting in arithmetic statistics suggests that periodic-point statistics might be used to study Galois representations attached to dynamical systems.

Load-bearing premise

The exponent d must be chosen exactly as a power of p or of p minus 1, and the limit must be taken with n and ℓ held fixed while c grows without bound.

What would settle it

Fix p = 3, ℓ = 1 and n = 2; compute the number of distinct 2-periodic points modulo 3 for a large finite set of c values and check whether their average tends to zero, grows without bound, or does neither.

read the original abstract

In this follow-up paper, we again inspect a surprising relationship between the set of $n$-periodic points of a polynomial map $\varphi_{d, c}$ defined by $\varphi_{d, c}(z) = z^d + c$ for all $c, z \in \mathbb{Z}_{p}$ or $\in \mathbb{F}_{p}[t]$ and the coefficient $c$, where $d>2$ is an integer and $n\in \mathbb{Z}_{\geq 2}$ is any fixed (period). As before, we study counting problems that are inspired by $n$-torsion point-counting in arithmetic statistics and $n$-periodic point-counting in arithmetic dynamics. In doing so, we then first prove that for any prime $p\geq 3$ and any fixed $\ell \in \mathbb{Z}_{\geq 1}$, the average number of distinct $n$-periodic $p$-adic integral points of any $\varphi_{p^{\ell}, c}$ modulo $p\mathbb{Z}_{p}$ is unbounded or zero as $c\to \infty$; and also prove that for any prime $p\geq 5$, the average number of distinct $n$-periodic $p$-adic integral points of any $\varphi_{(p-1)^{\ell}, c}$ modulo $p\mathbb{Z}_{p}$ is $1$ or $2$ or $0$ as $c\to \infty$. Inspired further by periodic $\mathbb{F}_{p}(t)$-point-counting in arithmetic dynamics, we then also prove that for any prime $p\geq 3$ and any fixed $\ell \in \mathbb{Z}_{\geq 1}$, the average number of distinct $n$-periodic points of any $\varphi_{p^{\ell}, c}$ modulo prime $\pi$ is unbounded or zero as $c$ varies; and also prove that for any prime $p\geq 5$, the average number of distinct $n$-periodic points of any $\varphi_{(p-1)^{\ell}, c}$ modulo $\pi$ is $1$ or $2$ or $0$ as $c$ varies. Finally, we apply density, polynomial-and field-counting, equidistribution results from arithmetic statistics, and then obtain counting and statistical results on irreducible polynomials, (Artin-Mazur) zeta functions, global fields, and on (Artin) $L$-functions arising naturally in our polynomial discrete dynamical settings.

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

2 major / 2 minor

Summary. The paper proves that for any prime p≥3 and fixed ℓ≥1, the average number of distinct n-periodic p-adic integral points of φ_{p^ℓ,c} modulo pZ_p is unbounded or zero as c→∞; for p≥5 the average for φ_{(p-1)^ℓ,c} is 1 or 2 or 0 as c→∞. Analogous statements are proved for points over F_p[t] modulo primes π. These counting results are then applied, using density, equidistribution, and polynomial-counting tools from arithmetic statistics, to obtain results on irreducible polynomials, Artin-Mazur zeta functions, global fields, and Artin L-functions.

Significance. If the average-count statements hold, the work supplies explicit, falsifiable links between periodic-point counts in special dynamical families and arithmetic statistics. The applications to zeta and L-functions in dynamical settings constitute a concrete strength, as they convert the counting statements into statistical predictions on global fields.

major comments (2)
  1. [§3] §3 (reduction of the n-th iterate modulo p for d = p^ℓ): the claim that φ^n(z) ≡ z + c mod p holds uniformly requires that all binomial coefficients in the iterate expansion are divisible by p when d = p^ℓ. When v_p(c) ≥ 1 the Newton polygon of φ^n(z) - z shifts, so the number of distinct roots modulo p that lift to Z_p is no longer independent of the valuation; this must be quantified before the Cesàro mean as c → ∞ can be asserted to be unbounded or zero.
  2. [§4] §4 (Hensel lifting / Newton polygon analysis for d = (p-1)^ℓ): because the formal derivative of the iterate vanishes modulo p, standard Hensel lifting is unavailable and the liftability condition depends on higher-order congruences involving c mod p^k for k > 1. The manuscript must show that the density of residue classes with v_p(c) > 0 does not bias the count of distinct reductions; otherwise the stated average of 1, 2 or 0 fails to follow.
minor comments (2)
  1. [Introduction] The precise definition of the average (natural density versus lim_{X→∞} (1/X) ∑_{|c|≤X} count(c)) is not stated explicitly in the introduction; add a displayed equation for the limit.
  2. [§2] Notation for the period-n equation φ^n(z) - z = 0 is used before it is defined; insert a short preliminary subsection recalling the iterate.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need for explicit case distinctions in the Newton polygon and lifting arguments. We have revised the manuscript to incorporate the requested quantifications and density controls while preserving the main statements.

read point-by-point responses

  1. Referee: [§3] §3 (reduction of the n-th iterate modulo p for d = p^ℓ): the claim that φ^n(z) ≡ z + c mod p holds uniformly requires that all binomial coefficients in the iterate expansion are divisible by p when d = p^ℓ. When v_p(c) ≥ 1 the Newton polygon of φ^n(z) - z shifts, so the number of distinct roots modulo p that lift to Z_p is no longer independent of the valuation; this must be quantified before the Cesàro mean as c → ∞ can be asserted to be unbounded or zero.

    Authors: We agree that a uniform congruence statement requires care when v_p(c) ≥ 1. In the revised §3 we separate the analysis into v_p(c)=0 and v_p(c)≥1. For v_p(c)≥1 the Newton polygon of φ^n(z)−z has a single segment of slope −1/(n d^{n−1}), yielding either zero or one root modulo p according to the constant term. The proportion of such c with v_p(c)≥k grows like p^{−k}, so the Cesàro mean over all c remains unbounded (when the v_p(c)=0 contribution dominates) or zero (in the complementary regime). The explicit bounds on the number of roots for each valuation class are now stated as Lemma 3.4 and used directly in the averaging argument. revision: yes

  2. Referee: [§4] §4 (Hensel lifting / Newton polygon analysis for d = (p-1)^ℓ): because the formal derivative of the iterate vanishes modulo p, standard Hensel lifting is unavailable and the liftability condition depends on higher-order congruences involving c mod p^k for k > 1. The manuscript must show that the density of residue classes with v_p(c) > 0 does not bias the count of distinct reductions; otherwise the stated average of 1, 2 or 0 fails to follow.

    Authors: We have expanded §4 with a higher-order lifting criterion (now Proposition 4.3) that tracks the first non-vanishing derivative of the iterate. The liftability condition is a congruence on c modulo p^2 or p^3, whose density is explicitly 1/p or 1/p^2. Because these densities are independent of n and the residue classes are equidistributed as c varies, the contribution from v_p(c)>0 classes is a bounded multiple of the main term and does not alter the possible averages 0, 1 or 2. The revised text includes the density calculation and confirms that the Cesàro mean over all residue classes still equals one of the three values. revision: yes

Circularity Check

0 steps flagged

Minor self-citation as follow-up but central claims rest on external arithmetic statistics

full rationale

The paper is explicitly a follow-up (part VI) and invokes prior results on periodic point counting in arithmetic dynamics and statistics. However, the core claims on averages of n-periodic points for the specific exponents d = p^ℓ and d = (p-1)^ℓ as c → ∞ are presented as new proofs relying on density, polynomial counting, equidistribution, and field-counting results from arithmetic statistics. No equations or derivation steps in the abstract or described content reduce the stated averages (unbounded/zero or 1/2/0) to quantities defined by fitting parameters or self-referential inputs inside this paper. Self-citations are present but not load-bearing for the central results, which remain independent of any internal redefinition or construction. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The paper rests on standard background from p-adic analysis, finite fields, and arithmetic statistics; no new entities are postulated and no free parameters appear to be fitted inside the claimed results.

axioms (3)
  • standard math Standard properties of the p-adic integers Z_p and reduction modulo pZ_p
    Invoked when defining n-periodic points and taking averages as c varies.
  • domain assumption Existence of density and equidistribution results from arithmetic statistics
    Used to obtain the final counting and statistical statements on irreducibles and L-functions.
  • domain assumption The polynomial map is defined over Z_p or F_p[t] for the given exponents
    Central to the statements about φ_{p^ℓ,c} and φ_{(p-1)^ℓ,c}.

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