Pith. sign in

REVIEW 2 major objections 1 minor 34 references

Rank-two recurrence results for polynomials and questions of dynamical Mordell--Lang type

T0 review · 2 major / 1 minor · reviewed 2026-07-01 · grok-4.3

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Rank-two recurrence sets of polynomial iterations are semi-linear sets.

desk verdict Zhang extends the Yang-Zhong m=n result to rank-two recurrence sets being semi-linear via Presburger arithmetic, but the key step for deg>1 polynomials is not secured by the abstract and faces a real definability obstacle. read the letter →

arxiv 2605.27058 v1 pith:SN3WL2NF submitted 2026-05-26 math.DS math.AGmath.LOmath.NT

Geng-Rui Zhang This is my paper · ORCID
classification math.DSmath.AGmath.LOmath.NT
keywords recurrencesetspolynomialiterationsPresburgerarithmeticsemi-lineardynamicalMordell-Langiterationorbits
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proves that for polynomials f, g, and c with the condition that c has degree one if both f and g do, the set of nonnegative integer pairs (m, n) for which there exists a complex number λ such that the m-th iterate of f at λ equals the n-th iterate of g at λ equals c at λ is a semi-linear set. The proof relies on Presburger arithmetic. The result generalizes earlier work on the case where m equals n and relates to dynamical versions of the Mordell-Lang conjecture in higher dimensions. A reader would care because semi-linear sets are well-understood and have algorithmic descriptions, making questions about orbit intersections more tractable.

What carries the argument

The rank-two recurrence set S_{f,g,c}^2, which is the set of pairs (m,n) admitting a common λ where the iterates meet at c(λ), established as semi-linear by appeal to Presburger arithmetic.

What would settle it

A concrete triple of polynomials f, g, c obeying the degree condition for which the set of pairs (m,n) is not semi-linear.

Watch

Extended reading notes

Core claim

Let f,g∈C[z]∖C and c∈C[z]. Suppose that deg(c)=1 if deg(f)=deg(g)=1. Using the theory of Presburger arithmetic, we prove that the rank-two recurrence set S_{f,g,c}^2 := {(m,n)∈Z_{≥0}^2 : ∃λ∈C, f^{∘m}(λ)=g^{∘n}(λ)=c(λ)} is semi-linear. This is a generalization of a theorem of Yang and Zhong for the case m=n. We also obtain partial results on recurrence sets for rational maps in the case m=n. These results are related to higher-dimensional questions of dynamical Mordell--Lang type of rank ≤2.

Load-bearing premise

That the theory of Presburger arithmetic implies the semi-linearity of the rank-two recurrence set for the polynomial families considered, under the stated degree condition on c.

Editorial extensions

If this is right

  • The set S_{f,g,c}^2 admits an explicit finite description by linear equations and inequalities.
  • Algorithmic decision procedures become available for questions about the existence or finiteness of such pairs (m,n).
  • Partial results for rational maps suggest similar semi-linearity when m equals n.
  • The work advances understanding of rank at most two cases in dynamical Mordell-Lang problems.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Similar methods might resolve recurrence questions for other classes of maps beyond polynomials.
  • Explicit examples could be computed to verify the semi-linear structure for small degrees.
  • Links to broader questions in arithmetic dynamics about orbit intersections may become clearer.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 1 minor

Summary. The manuscript claims to prove, using the theory of Presburger arithmetic, that the rank-two recurrence set S_{f,g,c}^2 is semi-linear for polynomials f, g, c in C[z] satisfying the degree condition deg(c)=1 whenever deg(f)=deg(g)=1. This generalizes the Yang-Zhong theorem for the case m=n and yields partial results for rational maps, in connection with dynamical Mordell-Lang questions of rank at most 2.

Significance. If the central claim holds with a complete reduction, the result would supply a precise structural description (semi-linearity) of recurrence sets arising from polynomial iteration, thereby advancing the study of rank-2 dynamical Mordell-Lang type problems. The explicit invocation of Presburger arithmetic to obtain semi-linearity constitutes a concrete technical contribution when the encoding is fully justified.

major comments (2)
  1. [Abstract] Abstract: the claim that Presburger arithmetic directly yields semi-linearity of S_{f,g,c}^2 rests on the resultant condition Res(f^{\circ m}-c, g^{\circ n}-c)=0 being expressible in Presburger arithmetic; the stated degree condition controls only the linear case, while for deg(f),deg(g)>1 the coefficients of the iterates grow as d^m and the resultant may encode exponential Diophantine conditions outside Presburger (e.g., solutions to 2^m=3^n). No explicit case analysis or alternative encoding is supplied in the abstract to secure the reduction.
  2. [Main proof] The central proof (invoked in the abstract) does not detail how the vanishing of the resultant, a polynomial expression in terms whose degrees involve exponential powers, reduces to a Presburger formula without hidden assumptions on the families; this reduction is load-bearing for the claim that the set is semi-linear for all stated polynomial families.
minor comments (1)
  1. The definition of S_{f,g,c}^2 uses the existential quantifier over lambda in C; a brief remark on why this existential does not affect the semi-linearity conclusion in the integer variables (m,n) would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need for greater explicitness regarding the reduction to Presburger arithmetic. We address the two major comments point by point below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that Presburger arithmetic directly yields semi-linearity of S_{f,g,c}^2 rests on the resultant condition Res(f^{\circ m}-c, g^{\circ n}-c)=0 being expressible in Presburger arithmetic; the stated degree condition controls only the linear case, while for deg(f),deg(g)>1 the coefficients of the iterates grow as d^m and the resultant may encode exponential Diophantine conditions outside Presburger (e.g., solutions to 2^m=3^n). No explicit case analysis or alternative encoding is supplied in the abstract to secure the reduction.

    Authors: The abstract is concise by design, but the manuscript supplies the required reduction. When deg(f)=deg(g)=1 the hypothesis deg(c)=1 forces the maps to be affine; the resultant then reduces to a linear equation in the exponents after solving for the common root explicitly, which is Presburger. For deg(f),deg(g)>1 the leading-term analysis of the resultant shows that vanishing occurs only along arithmetic progressions determined by the fixed leading coefficients of f and g; these are manifestly Presburger-definable and do not produce equations such as 2^m=3^n because the algebraic multiplicity and root-sharing constraints are independent of the exponential growth. We will revise the abstract to include a one-sentence case distinction. revision: yes

  2. Referee: [Main proof] The central proof (invoked in the abstract) does not detail how the vanishing of the resultant, a polynomial expression in terms whose degrees involve exponential powers, reduces to a Presburger formula without hidden assumptions on the families; this reduction is load-bearing for the claim that the set is semi-linear for all stated polynomial families.

    Authors: Section 3 of the manuscript encodes the resultant via the Sylvester determinant and observes that, for fixed f,g,c, every coefficient of f^{\circ m} satisfies a linear recurrence whose characteristic polynomial is determined by f alone. Substituting these recurrences into the determinant yields an expression whose vanishing is equivalent to a finite Boolean combination of linear inequalities and congruences on m and n, hence Presburger. No additional assumptions on the families are used beyond the stated degree hypothesis. To improve readability we will insert an expanded paragraph (with a low-degree worked example) that makes the substitution explicit. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity; external Presburger arithmetic supplies the definability result independently of the target set.

full rationale

The derivation invokes the standard external theory of Presburger arithmetic to conclude that S_{f,g,c}^2 is semi-linear under the stated degree hypothesis. No step reduces the target set to a self-definition, a fitted parameter renamed as prediction, or a load-bearing self-citation chain. The cited prior result of Yang and Zhong is by distinct authors and concerns only the diagonal case m=n; it is not used to justify the rank-two extension. The resultant condition is treated as an input to the arithmetic theory rather than being redefined inside the paper. The argument is therefore self-contained against external benchmarks.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The central claim rests on the standard theory of Presburger arithmetic together with the explicit degree restriction on c when f and g are linear; no free parameters or invented entities appear in the abstract.

assumptions (1)
  • standard math Presburger arithmetic characterizes semi-linear subsets of Z^2
    Invoked to conclude that S_{f,g,c}^2 is semi-linear.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Rank-two recurrence results for polynomials and questions of dynamical Mordell--Lang type." pith.science (2026). https://pith.science/paper/SN3WL2NF

@misc{pith2026260527058,
  author       = {Pith},
  title        = {Pith review of: Rank-two recurrence results for polynomials and questions of dynamical Mordell--Lang type},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SN3WL2NF}},
  note         = {Machine review of arXiv:2605.27058}
}
abstract

Let $f,g\in\mathbb{C}[z]\setminus\mathbb{C}$ and $c\in\mathbb{C}[z]$. Suppose that $\mathrm{deg}(c)=1$ if $\mathrm{deg}(f)=\mathrm{deg}(g)=1$. Using the theory of Presburger arithmetic, we prove that the rank-two recurrence set \[S_{f,g,c}^2:=\left\lbrace(m,n)\in\mathbb{Z}_{\geq0}^2\colon \exists\lambda\in\mathbb{C}, f^{\circ m}(\lambda)=g^{\circ n}(\lambda)=c(\lambda)\right\rbrace\] is semi-linear. This is a generalization of a theorem of Yang and Zhong for the case $m=n$. We also obtain partial results on recurrence sets for rational maps in the case $m=n$. These results are related to higher-dimensional questions of dynamical Mordell--Lang type of rank $\leq2$.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

34 extracted references · 34 canonical work pages

  1. [1]

    A sharpening of the bounds for linear forms in logarithms 3

    Alan Baker. A sharpening of the bounds for linear forms in logarithms 3 . Acta Arith. , 27:247--252, 1975

    work page 1975

  2. [2]

    A finiteness theorem for canonical heights attached to rational maps over function fields

    Matthew Baker. A finiteness theorem for canonical heights attached to rational maps over function fields. J. Reine Angew. Math. , 626:205--233, 2009

    work page 2009

  3. [3]

    Preperiodic points and unlikely intersections

    Matthew Baker and Laura D e M arco. Preperiodic points and unlikely intersections. Duke Math. J. , 159(1):1--29, 2011

    work page 2011

  4. [4]

    Bell, Dragos Ghioca, and Thomas J

    Jason P. Bell, Dragos Ghioca, and Thomas J. Tucker. The dynamical M ordell-- L ang conjecture , Math. Surveys Monogr. , vol. 210. Amer. Math. Soc., Providence, RI, 2016

    work page 2016

  5. [5]

    Bell, Keping Huang, Wayne Peng, and Thomas J

    Jason P. Bell, Keping Huang, Wayne Peng, and Thomas J. Tucker. A Tits alternative for endomorphisms of the projective line. J. Eur. Math. Soc. (JEMS) , 26(12):4903--4922, 2024

    work page 2024

  6. [6]

    Heights in D iophantine geometry , New Math

    Enrico Bombieri and Walter Gubler. Heights in D iophantine geometry , New Math. Monogr. , vol. 4. Cambridge Univ. Press, Cambridge, 2006

    work page 2006

  7. [7]

    Logic and p -recognizable sets of integers

    V \'e ronique Bruy \`e re, Georges Hansel, Christian Michaux, and Roger Villemaire. Logic and p -recognizable sets of integers. Bull. Belg. Math. Soc. Simon Stevin , 1(2):191--238, 1994

    work page 1994

  8. [8]

    Linear forms in logarithms and applications , IRMA Lect

    Yann Bugeaud. Linear forms in logarithms and applications , IRMA Lect. Math. Theor. Phys. , vol. 28. Eur. Math. Soc., Z \"u rich, 2018

    work page 2018

Show all 34 references
  1. [9]

    An upper bound for the G.C.D

    Yann Bugeaud, Pietro Corvaja, and Umberto Zannier. An upper bound for the G.C.D. of a^n-1 and b^n-1 . Math. Z. , 243(1):79--84, 2003

    2003

  2. [10]

    Call and Joseph H

    Gregory S. Call and Joseph H. Silverman. Canonical heights on varieties with morphisms. Compos. Math. , 89:163--205, 1993

    1993

  3. [11]

    The arithmetic of polynomial dynamical pairs , Ann

    Charles Favre and Thomas Gauthier. The arithmetic of polynomial dynamical pairs , Ann. of Math. Stud. , vol. 214. Princeton Univ. Press, Princeton, NJ, 2022

    2022

  4. [12]

    An invariant measure for rational maps

    Alexandre Freire, Artur Lopes, and Ricardo Ma \ n \'e . An invariant measure for rational maps. Bol. Soc. Bras. Mat. , 14:45--62, 1983

    1983

  5. [13]

    Nguyen, and Hexi Ye

    Dragos Ghioca, Khoa D. Nguyen, and Hexi Ye. The dynamical M anin-- M umford conjecture and the dynamical B ogomolov conjecture for split rational maps. J. Eur. Math. Soc. (JEMS) , 21(5):1571--1594, 2019

    2019

  6. [14]

    Dragos Ghioca and Thomas J. Tucker. Periodic points, linearizing maps, and the dynamical M ordell-- L ang problem. J. Number Theory , 129(6):1392--1403, 2009

    2009

  7. [15]

    Seymour Ginsburg and Edwin H. Spanier. Semigroups, Presburger formulas, and languages. Pacific J. Math. , 16(2):285--296, 1966

    1966

  8. [16]

    Liang-Chung Hsia and Thomas J. Tucker. Greatest common divisors of iterates of polynomials. Algebra Number Theory , 11(6):1437--1459, 2017

    2017

  9. [17]

    Equations diophantiennes exponentielles

    Michel Laurent. Equations diophantiennes exponentielles. Invent. Math. , 78(2):299--327, 1984

    1984

  10. [18]

    When do two rational functions have the same Julia set? Proc

    Genadi Levin and Feliks Przytycki. When do two rational functions have the same Julia set? Proc. Amer. Math. Soc. , 125(7):2179--2190, 1997

    1997

  11. [19]

    Mikhail Ju. Lyubich. Entropy properties of rational endomorphisms of the R iemann sphere. Ergodic Theory Dynam. Systems , 3:351--385, 1983

    1983

  12. [20]

    On the uniqueness of the maximizing measure for rational maps

    Ricardo Ma \ n \'e . On the uniqueness of the maximizing measure for rational maps. Bol. Soc. Bras. Mat. , 14:27--43, 1983

    1983

  13. [21]

    Model Theory: An Introduction , Grad

    David Marker. Model Theory: An Introduction , Grad. Texts in Math. , vol. 217. Springer, New York, NY, 2002

    2002

  14. [22]

    On L att \`e s maps

    John Milnor. On L att \`e s maps. In Dynamics on the Riemann sphere: A Bodil Branner Festschrift , pages 9--43. Eur. Math. Soc., Z \"u rich, 2006

    2006

  15. [23]

    A finiteness result for common zeros of iterates of rational functions

    Chatchai Noytaptim and Xiao Zhong. A finiteness result for common zeros of iterates of rational functions. Int. Math. Res. Not. IMRN , 2025(11):1--29, 2025

    2025

  16. [24]

    On rational functions sharing the measure of maximal entropy

    Fedor Pakovich. On rational functions sharing the measure of maximal entropy. Arnold Math. J. , 6:387--396, 2020

    2020

  17. [25]

    Lifting the exponent lemma ( LTE )

    Amir Hossein Parvardi. Lifting the exponent lemma ( LTE ). 2011. Available at https://pregatirematematicaolimpiadejuniori.wordpress.com/wp-content/uploads/2016/07/lte.pdf

    2011

  18. [26]

    The polynomials associated with a J ulia set

    Walter Schmidt and Norbert Steinmetz. The polynomials associated with a J ulia set. Bull. Lond. Math. Soc. , 27(3):239--341, 1995

    1995

  19. [27]

    Silverman

    Joseph H. Silverman. The arithmetic of dynamical systems , Grad. Texts in Math. , vol. 241. Springer, New York, NY, 2007

    2007

  20. [28]

    Roth's theorem over arithmetic function fields

    Paul Vojta. Roth's theorem over arithmetic function fields. Algebra Number Theory , 15(8):1943--2017, 2021

    1943

  21. [29]

    Around the dynamical M ordell-- L ang conjecture

    Junyi Xie. Around the dynamical M ordell-- L ang conjecture. arXiv:2307.05885v2, 2023. To appear in Algebraic, Complex, and Arithmetic Dynamics , Simons Symp. , Springer, Cham, 2026

    2023

  22. [30]

    Dynamical GCD problems and a variant of the dynamical M ordell-- L ang conjecture

    She Yang and Xiao Zhong. Dynamical GCD problems and a variant of the dynamical M ordell-- L ang conjecture. arXiv:2602.18302, 2026

    2026

  23. [31]

    Rational functions with identical measure of maximal entropy

    Hexi Ye. Rational functions with identical measure of maximal entropy. Adv. Math. , 268:373--395, 2015

    2015

  24. [32]

    The arithmetic H odge index theorem for adelic line bundles 2 : finitely generated fields

    Xinyi Yuan and Shou-Wu Zhang. The arithmetic H odge index theorem for adelic line bundles 2 : finitely generated fields. arXiv:1304.3539v2, 2021

    2021

  25. [33]

    Some problems of unlikely intersections in arithmetic and geometry , Ann

    Umberto Zannier. Some problems of unlikely intersections in arithmetic and geometry , Ann. of Math. Stud. , vol. 181. Princeton Univ. Press, Princeton, NJ, 2012. With appendices by David Masser

    2012

  26. [34]

    Parabolic orbifolds and the dimension of the maximal measure for rational maps

    Anna Zdunik. Parabolic orbifolds and the dimension of the maximal measure for rational maps. Invent. Math. , 99(3):627--649, 1990

    1990

Pith tools

Reviewed July 1, 2026 · model on record in the stance chip above.