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arxiv: 2604.08795 · v1 · submitted 2026-04-09 · 🧮 math.NT · math.DS

A Dynamical Lifting Problem For Additive Polynomials

Daniel Tedeschi This is my paper

Pith reviewed 2026-05-10 16:50 UTC · model grok-4.3

classification 🧮 math.NT math.DS
keywords additive polynomialsGalois coverslifting problemdynamical systemsfinite fieldslinear conjugacy classesseparable polynomialsalgebraic curves
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The pith

Additive separable polynomials over the algebraic closure of F_p admit no solutions to the dynamical analogue of the Galois lifting problem.

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

The paper defines a dynamical version of the classical lifting problem for Galois covers of algebraic curves. It then proves that this problem has no solutions when restricted to additive and separable polynomials over the algebraic closure of a finite field. The work also determines the exact dimension of the space of linear conjugacy classes in the matrix ring M_{p^m} that contain at least one such polynomial. A sympathetic reader sees this as establishing a clear negative result that distinguishes dynamical behavior from its static algebraic counterpart.

Core claim

A dynamical analogue of the lifting problem for Galois covers is introduced and shown to have a negative solution for the collection of additive separable polynomials over the algebraic closure of F_p. The dimension of the space of linear conjugacy classes in M_{p^m} over the same field that contain an additive separable polynomial is computed explicitly.

What carries the argument

The dynamical lifting problem, defined as an analogue of the classical Galois-cover lifting problem and applied to additive separable polynomials.

Load-bearing premise

The definition of the dynamical lifting problem faithfully captures the intended analogue of the classical Galois-cover lifting problem.

What would settle it

An explicit additive separable polynomial over the algebraic closure of F_p together with a lift that satisfies the dynamical condition, or an independent calculation showing a different dimension for the space of linear conjugacy classes.

Figures

Figures reproduced from arXiv: 2604.08795 by Daniel Tedeschi.

Figure 1
Figure 1. Figure 1: Algebraic and Geometric Perspectives Let Pf denote the image of Pf under the canonical map P 1 k → P 1 k , which for the affine patch k[t] takes a prime ideal p to its intersection p∩k[t]. Standard algebraic number theory results imply that Pf contains the branch locus of each iterate f n . Even stronger, P 1 k \ Pf is the maximal subset of P 1 k over which every iterate of f is ´etale. We say that f is po… view at source ↗
read the original abstract

We introduce a dynamical analogue of the lifting problem for Galois covers of algebraic curves and find a negative solution for the collection of additive, separable polynomials over $\overline{\mathbb{F}}_p$. We also explicitly compute the dimension of the space of linear conjugacy classes in $M_{p^m}(\overline{\mathbb{F}}_p)$ which contain an additive, separable polynomial.

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

1 major / 1 minor

Summary. The manuscript introduces a dynamical analogue of the lifting problem for Galois covers of algebraic curves. For additive separable polynomials over the algebraic closure of F_p it establishes a negative result (no such lifts exist) and explicitly computes the dimension of the space of linear conjugacy classes in M_{p^m}(F_p-bar) containing an additive separable polynomial.

Significance. If the dynamical analogue is well-motivated, the negative result supplies a concrete obstruction for this family of maps and the dimension formula quantifies the conjugacy classes in a computable way; both could serve as test cases or building blocks for further work in arithmetic dynamics over finite fields.

major comments (1)
  1. [Definition of dynamical lifting problem] The section defining the dynamical lifting problem: the negative solution is load-bearing only if this definition imposes obstructions equivalent to those in the classical Galois-cover lifting problem (prescribed Galois action or ramification data). The manuscript should include an explicit comparison or lemma showing that the dynamical version (presumably via iteration or functional equations) faithfully translates the classical conditions; otherwise the claimed negative answer addresses a different question.
minor comments (1)
  1. [Abstract] The abstract states the dimension result but does not indicate the method (e.g., whether it proceeds by counting fixed points of the conjugation action or by direct linear-algebraic arguments); a one-sentence hint would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and valuable feedback on our manuscript. We agree that clarifying the relationship between the dynamical lifting problem and the classical Galois-cover lifting problem will strengthen the paper. We address the major comment below and will incorporate the suggested revision.

read point-by-point responses

  1. Referee: The section defining the dynamical lifting problem: the negative solution is load-bearing only if this definition imposes obstructions equivalent to those in the classical Galois-cover lifting problem (prescribed Galois action or ramification data). The manuscript should include an explicit comparison or lemma showing that the dynamical version (presumably via iteration or functional equations) faithfully translates the classical conditions; otherwise the claimed negative answer addresses a different question.

    Authors: We agree that an explicit comparison is needed to confirm that the dynamical formulation faithfully captures the classical obstructions. In the revised version, we will add a lemma (or subsection) immediately following the definition of the dynamical lifting problem. This lemma will explicitly relate the functional equation satisfied by a lift to the preservation of prescribed Galois action and ramification data under iteration, showing that our negative result for additive separable polynomials provides a genuine obstruction in the classical sense for this family. We believe this addition will address the concern without altering the main results. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation introduces new definition and states independent negative result

full rationale

The paper introduces a dynamical analogue of the classical lifting problem and asserts a negative solution for additive separable polynomials over the algebraic closure of F_p, together with an explicit dimension computation for linear conjugacy classes. No equations, self-definitions, fitted parameters renamed as predictions, or load-bearing self-citations appear in the provided abstract or description that would reduce the claimed result to its own inputs by construction. The central claims are presented as direct mathematical statements rather than tautological rephrasings of definitions or prior self-citations.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are identifiable from the abstract alone.

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Reference graph

Works this paper leans on

31 extracted references · 31 canonical work pages

  1. [1]

    Profinite iterated monodromy groups of unicritical polynomials, 2025

    Ophelia Adams and Trevor Hyde. Profinite iterated monodromy groups of unicritical polynomials, 2025

    work page 2025

  2. [2]

    Local arboreal representa- tions.Int

    Jacqueline Anderson, Spencer Hamblen, Bjorn Poonen, and Laura Walton. Local arboreal representa- tions.Int. Math. Res. Not. IMRN, (19):5974–5994, 2018

    work page 2018

  3. [3]

    Benedetto, Xander Faber, Benjamin Hutz, Jamie Juul, and Yu Yasufuku

    Robert L. Benedetto, Xander Faber, Benjamin Hutz, Jamie Juul, and Yu Yasufuku. A large arboreal Galois representation for a cubic postcritically finite polynomial.Res. Number Theory, 3:Paper No. 29, 21, 2017

    work page 2017

  4. [4]

    Benedetto, Dragos Ghioca, Jamie Juul, and Thomas J

    Robert L. Benedetto, Dragos Ghioca, Jamie Juul, and Thomas J. Tucker. Specializations of iterated Galois groups of PCF rational functions.Math. Ann., 392(1):1031–1050, 2025

    work page 2025

  5. [5]

    Bouw, ¨Ozlem Ejder, and Valentijn Karemaker

    Irene I. Bouw, ¨Ozlem Ejder, and Valentijn Karemaker. Dynamical Belyi maps and arboreal Galois groups.Manuscripta Math., 165(1-2):1–34, 2021

    work page 2021

  6. [6]

    A census of rational maps.Conform

    Eva Brezin, Rosemary Byrne, Joshua Levy, Kevin Pilgrim, and Kelly Plummer. A census of rational maps.Conform. Geom. Dyn., 4:35–74, 2000

    work page 2000

  7. [7]

    Adrien Douady and John H. Hubbard. A proof of Thurston’s topological characterization of rational functions.Acta Math., 171(2):263–297, 1993

    work page 1993

  8. [8]

    Rational functions with a unique critical point.Int

    Xander Faber. Rational functions with a unique critical point.Int. Math. Res. Not. IMRN, (3):681–699, 2014

    work page 2014

  9. [9]

    On the dynamical galois group of certain affine polynomials in positive characteristic, 2026

    Andrea Ferraguti and Guido Maria Lido. On the dynamical galois group of certain affine polynomials in positive characteristic, 2026

    work page 2026

  10. [10]

    Springer-Verlag, Berlin, 1996

    David Goss.Basic structures of function field arithmetic, volume 35 ofErgebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Springer-Verlag, Berlin, 1996

    work page 1996

  11. [11]

    Liftings of Galois covers of smooth curves.Compositio Math., 113(3):237–272, 1998

    Barry Green and Michel Matignon. Liftings of Galois covers of smooth curves.Compositio Math., 113(3):237–272, 1998

    work page 1998

  12. [12]

    Patching and Galois theory

    David Harbater. Patching and Galois theory. InGalois groups and fundamental groups, volume 41 of Math. Sci. Res. Inst. Publ., pages 313–424. Cambridge Univ. Press, Cambridge, 2003

    work page 2003

  13. [13]

    Abhyankar’s conjectures in Galois theory: current status and future directions.Bull

    David Harbater, Andrew Obus, Rachel Pries, and Katherine Stevenson. Abhyankar’s conjectures in Galois theory: current status and future directions.Bull. Amer. Math. Soc. (N.S.), 55(2):239–287, 2018

    work page 2018

  14. [14]

    Th´ eorie des Nombres et Applications

    Rafe Jones. Galois representations from pre-image trees: an arboreal survey. InActes de la Conf´ erence “Th´ eorie des Nombres et Applications”, volume 2013 ofPubl. Math. Besan¸ con Alg` ebre Th´ eorie Nr., pages 107–136. Presses Univ. Franche-Comt´ e, Besan¸ con, 2013

    work page 2013

  15. [15]

    Local fields, iterated extensions, and julia sets, 2025

    Pui Hang Lee, Michelle Manes, and Nha Xuan Truong. Local fields, iterated extensions, and julia sets, 2025

    work page 2025

  16. [16]

    Hyperbolic components

    John Milnor. Hyperbolic components. InConformal dynamics and hyperbolic geometry, volume 573 of Contemp. Math., pages 183–232. Amer. Math. Soc., Providence, RI, 2012. With an appendix by A. Poirier. 16 D. TEDESCHI

    work page 2012

  17. [17]

    Iterated monodromy groups

    Volodymyr Nekrashevych. Iterated monodromy groups. InGroups St Andrews 2009 in Bath. Volume 1, volume 387 ofLondon Math. Soc. Lecture Note Ser., pages 41–93. Cambridge Univ. Press, Cambridge, 2011

    work page 2009

  18. [18]

    Springer-Verlag, Berlin, 1999

    J¨ urgen Neukirch.Algebraic number theory, volume 322 ofGrundlehren der mathematischen Wis- senschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1999. Trans- lated from the 1992 German original and with a note by Norbert Schappacher, With a foreword by G. Harder

    work page 1999

  19. [19]

    The (local) lifting problem for curves

    Andrew Obus. The (local) lifting problem for curves. InGalois-Teichm¨ uller theory and arithmetic ge- ometry, volume 63 ofAdv. Stud. Pure Math., pages 359–412. Math. Soc. Japan, Tokyo, 2012

    work page 2012

  20. [20]

    Cyclic extensions and the local lifting problem.Ann

    Andrew Obus and Stefan Wewers. Cyclic extensions and the local lifting problem.Ann. of Math. (2), 180(1):233–284, 2014

    work page 2014

  21. [21]

    R. W. K. Odoni. The Galois theory of iterates and composites of polynomials.Proc. London Math. Soc. (3), 51(3):385–414, 1985

    work page 1985

  22. [22]

    R. W. K. Odoni. On the prime divisors of the sequencew n+1 = 1 +w 1 · · ·w n.J. London Math. Soc. (2), 32(1):1–11, 1985

    work page 1985

  23. [23]

    R. W. K. Odoni. Realising wreath products of cyclic groups as Galois groups.Mathematika, 35(1):101– 113, 1988

    work page 1988

  24. [24]

    Finiteness and liftability of postcritically finite quadratic morphisms in arbitrary charac- teristic, 2013

    Richard Pink. Finiteness and liftability of postcritically finite quadratic morphisms in arbitrary charac- teristic, 2013

    work page 2013

  25. [25]

    The Oort conjecture on lifting covers of curves.Ann

    Florian Pop. The Oort conjecture on lifting covers of curves.Ann. of Math. (2), 180(1):285–322, 2014

    work page 2014

  26. [26]

    Silverman

    Joseph H. Silverman. The space of rational maps onP 1.Duke Math. J., 94(1):41–77, 1998

    work page 1998

  27. [27]

    Silverman.The arithmetic of dynamical systems

    Joseph H. Silverman.The arithmetic of dynamical systems. Springer, 2007

    work page 2007

  28. [28]

    The stacks project.https://stacks.math.columbia.edu, 2026

    The Stacks project authors. The stacks project.https://stacks.math.columbia.edu, 2026

    work page 2026

  29. [29]

    Cambridge University Press, Cambridge, 2009

    Tam´ as Szamuely.Galois groups and fundamental groups, volume 117 ofCambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2009

    work page 2009

  30. [30]

    B. L. van der Waerden. Die Zerlegungs-und Tr¨ agheitsgruppe als Permutationsgruppen.Math. Ann., 111(1):731–733, 1935

    work page 1935

  31. [31]

    Washington.Introduction to cyclotomic fields, volume 83 ofGraduate Texts in Mathematics

    Lawrence C. Washington.Introduction to cyclotomic fields, volume 83 ofGraduate Texts in Mathematics. Springer-Verlag, New York, 1982

    work page 1982