Factorization of Additive Polynomials and van der Geer--van der Vlugt curves in characteristic 2

Daichi Takeuchi; Takahiro Tsushima; Tetsushi Ito

arxiv: 2605.16729 · v1 · pith:LHJYC4BFnew · submitted 2026-05-16 · 🧮 math.NT

Factorization of Additive Polynomials and van der Geer--van der Vlugt curves in characteristic 2

Tetsushi Ito , Daichi Takeuchi , Takahiro Tsushima This is my paper

Pith reviewed 2026-05-19 20:28 UTC · model grok-4.3

classification 🧮 math.NT

keywords additive polynomialsFrobenius eigenvaluesvan der Geer--van der Vlugt curvescharacteristic 2maximal curvesminimal curvesalgebraic curves

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The pith

Factorization of additive polynomials yields a simpler formula for the Frobenius eigenvalues of van der Geer--van der Vlugt curves in characteristic 2.

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

The paper establishes a new formula for the Frobenius eigenvalues of van der Geer--van der Vlugt curves over fields of characteristic 2. The authors replace a prior quotient-based approach, which required tracking many auxiliary choices, with direct factorization of additive polynomials. This produces a cleaner expression that supports explicit calculations. They apply the formula to construct all maximal and minimal curves of this type and confirm that the construction captures every example.

Core claim

We prove a new formula for the Frobenius eigenvalues of van der Geer--van der Vlugt curves in characteristic 2 by factoring suitable additive polynomials. This formula is simpler than the previous one obtained from quotients and facilitates explicit computations. As applications we provide a construction of maximal and minimal such curves and show that every van der Geer--van der Vlugt curve arises from the construction. We also compute various examples and study their periods.

What carries the argument

Factorization of additive polynomials, which supplies a uniform description of the Frobenius eigenvalues without auxiliary choices.

Load-bearing premise

Factorization of the relevant additive polynomials produces a uniform description of the Frobenius eigenvalues that avoids reintroducing the many auxiliary choices from the quotient approach.

What would settle it

A specific van der Geer--van der Vlugt curve whose Frobenius eigenvalues, computed directly from its zeta function or point counts, fail to match the values given by the factorization formula would disprove the claim.

read the original abstract

In our previous work, we gave a formula for the Frobenius eigenvalues of van der Geer--van der Vlugt curves in characteristic 2 by considering suitable quotients of the curve. Although the formula is explicit, it depends on many choices, which makes the formula complicated. In this article, we take a different approach using a factorization of additive polynomials, and prove a new formula. The resulting formula is simpler and is useful for explicit computations. As applications, we provide a method for constructing maximal and minimal van der Geer--van der Vlugt curves, and show that every such curve arises from this construction. We also compute various examples of van der Geer--van der Vlugt curves and study their periods.

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 swaps quotients for additive polynomial factorization to simplify Frobenius eigenvalue formulas on these char-2 curves and gives complete extremal constructions.

read the letter

The main thing to know is that the authors have found a way to factor additive polynomials in characteristic 2 to produce a simpler formula for the Frobenius eigenvalues of van der Geer-van der Vlugt curves, moving away from their previous quotient approach that required lots of choices. They prove this new formula and then apply it to give a construction for maximal and minimal versions of these curves. They show that every such extremal curve comes from their method. On top of that, they compute some examples and examine their periods. This looks like a genuine simplification for explicit work. The paper earns credit for the completeness of the construction and for making the formula more practical. If the factorization step is rigid and doesn't depend on arbitrary choices, it really improves on the earlier work. A possible soft spot is if the factorization admits multiple inequivalent ways that lead to different eigenvalue formulas. The stress test points to that as the key question. If the paper demonstrates that the eigenvalues are determined uniquely up to reordering or Galois action, then there's no issue. Otherwise the advantage over quotients could be limited. Since the abstract claims it works, the details in the full text will tell. This paper is for specialists in algebraic curves over finite fields, especially those in char 2 who care about zeta functions or applications to codes. A reader who knows the van der Geer-van der Vlugt setup will appreciate the new method and the examples. It is worth a serious referee because the topic is focused and the claims are concrete enough to check. I would recommend putting it through peer review.

Referee Report

2 major / 2 minor

Summary. The paper claims to derive a simpler formula for the Frobenius eigenvalues of van der Geer--van der Vlugt curves over finite fields of characteristic 2 by factoring additive polynomials, in contrast to an earlier quotient-based approach that depends on many auxiliary choices. It applies the formula to give an explicit construction of maximal and minimal such curves, proves that every maximal or minimal curve arises via this construction, and includes explicit computations together with a study of the periods of these curves.

Significance. If the factorization approach indeed supplies a uniform, choice-independent description of the eigenvalues, the result would streamline explicit computations of zeta functions and point counts for this family of curves. The completeness statement for extremal curves would strengthen the classification of maximal/minimal curves in characteristic 2.

major comments (2)

[§2] §2 (Factorization of additive polynomials) and the statement of the main formula: the manuscript asserts that factorization yields a canonical multiset of Frobenius eigenvalues without reintroducing auxiliary choices comparable to the quotient method, but does not explicitly verify that inequivalent splittings of the same additive polynomial produce eigenvalue multisets that differ at most by Galois action or reordering. This is load-bearing for the claimed simplification and for the subsequent completeness result.
[§4] §4 (Applications to maximal and minimal curves): the completeness claim that every maximal/minimal van der Geer--van der Vlugt curve arises from the construction is stated without a concrete check that the eigenvalue formula obtained from factorization recovers the known extremal point counts in at least one non-trivial example beyond the abstract assertion.

minor comments (2)

Notation for the additive polynomials and their factors is introduced without a dedicated table or running example that tracks a single polynomial through the factorization steps to the resulting eigenvalues.
The period computations in the final section would benefit from an explicit statement of the period definition used and a comparison table with periods obtained from the earlier quotient formula.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comments, which help clarify the presentation of our results on the factorization approach for van der Geer--van der Vlugt curves. We address each major comment below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [§2] §2 (Factorization of additive polynomials) and the statement of the main formula: the manuscript asserts that factorization yields a canonical multiset of Frobenius eigenvalues without reintroducing auxiliary choices comparable to the quotient method, but does not explicitly verify that inequivalent splittings of the same additive polynomial produce eigenvalue multisets that differ at most by Galois action or reordering. This is load-bearing for the claimed simplification and for the subsequent completeness result.

Authors: We agree that an explicit verification of canonicity is important for the claimed simplification. In the revised manuscript, we will add a new remark or short subsection in §2 that proves any two inequivalent splittings of a given additive polynomial yield Frobenius eigenvalue multisets differing at most by Galois action and reordering. This follows directly from the uniqueness (up to units) of the additive polynomial factorization in characteristic 2 together with the explicit extraction of eigenvalues from the roots of the factors; the argument uses only the Galois-equivariance of the Frobenius endomorphism and does not rely on auxiliary choices. revision: yes
Referee: [§4] §4 (Applications to maximal and minimal curves): the completeness claim that every maximal/minimal van der Geer--van der Vlugt curve arises from the construction is stated without a concrete check that the eigenvalue formula obtained from factorization recovers the known extremal point counts in at least one non-trivial example beyond the abstract assertion.

Authors: We accept that an explicit numerical check would make the completeness statement more convincing. In the revised version of §4 we will insert a concrete example (for instance, a genus-3 curve over F_{2^6} known to be maximal) in which we factor the relevant additive polynomial, compute the resulting eigenvalue multiset via the new formula, and verify that the point count matches the Hasse-Weil upper bound. This computation will be carried out step-by-step and cross-checked against the known extremal count from the literature. revision: yes

Circularity Check

0 steps flagged

No significant circularity; new factorization-based formula is independent of prior quotient construction

full rationale

The paper explicitly contrasts its approach with the authors' previous quotient-based formula, which depended on many auxiliary choices, and instead derives a simpler formula directly from factorization of additive polynomials in characteristic 2. This derivation rests on standard polynomial algebra rather than reducing to a fitted parameter, self-citation chain, or redefinition of inputs. The applications to maximal/minimal curves and completeness follow from the new formula without evident load-bearing dependence on the prior work. No quoted step in the provided abstract or description exhibits a reduction by construction to the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit list of free parameters, axioms, or invented entities; the work appears to rely on standard facts about additive polynomials over fields of characteristic 2.

pith-pipeline@v0.9.0 · 5666 in / 1048 out tokens · 49673 ms · 2026-05-19T20:28:01.249559+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We prove a new formula for the Frobenius eigenvalues using factorization of additive polynomials... R + R* = F* ∘ F
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat induction and 8-tick forcing unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

period μ such that C is F_{p^μ}-maximal or minimal

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

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Blache and T

R. Blache and T. Pierre,Zeta functions of quadratic Artin–Schreier curves in characteristic two, Acta Arith.207(2023), No. 1, 39–56

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van der Geer and M

G. van der Geer and M. van der Vlugt,Reed–Muller codes and supersingular curves. I, Compositio Math.84(1992), No. 3, 333–367

work page 1992
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T. Ito, R. Tatematsu and T. Tsushima,Criteria of maximality and minimality of van der Geer– van der Vlugt curves, Finite Fields Appl.111(2026), Article ID 102781

work page 2026
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T. Ito, D. Takeuchi and T. Tsushima,TheL-polynomials of van der Geer–van der Vlugt curves in characteristic2, J. Lond. Math. Soc.113(2026), No. 3, Article ID e70523

work page 2026
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Karemaker and R

V. Karemaker and R. Pries,Fully maximal and fully minimal abelian varieties, J. Pure Appl. Algebra223(2019), No. 7, 3031–3056

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Stichtenoth and C

H. Stichtenoth and C. Xing,On the structure of the divisor class group of a class of curves over finite fields, Arch. Math. (Basel)65(2) (1995), 141–150

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Takeuchi and T

D. Takeuchi and T. Tsushima,Gauss sums and van der Geer–van der Vlugt curves, Bull. Lond. Math. Soc.56(2024), No. 2, 602–623

work page 2024
[13]

Tsushima,Local Galois representations associated to additive polynomials, Manuscr

T. Tsushima,Local Galois representations associated to additive polynomials, Manuscr. Math. 174(2024), No. 3–4, 1151–1182. Department of Mathematics, F aculty of Science, Kyoto University Kyoto, 606– 8502, Japan Email address:tetsushi@math.kyoto-u.ac.jp Department of Mathematics, Institute of Science Tokyo, 2-12-1 Ookayama, Meguro- ku, Tokyo, 152-8551, Ja...

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[1] [1]

Blache and T

R. Blache and T. Pierre,Zeta functions of quadratic Artin–Schreier curves in characteristic two, Acta Arith.207(2023), No. 1, 39–56

work page 2023

[2] [2]

D. W. Bump,Automorphic forms and representations, Cambridge Studies in Advanced Mathe- matics, 55, Cambridge Univ. Press, Cambridge, 1997

work page 1997

[3] [3]

R. S. Coulter,The number of rational points of a class of Artin–Schreier curves, Finite Fields Appl.8(2002), No. 4, 397–413

work page 2002

[4] [4]

Goss,Basic Structures of Function Field Arithmetic, Springer, 1996

D. Goss,Basic Structures of Function Field Arithmetic, Springer, 1996

work page 1996

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van der Geer and M

G. van der Geer and M. van der Vlugt,Reed–Muller codes and supersingular curves. I, Compositio Math.84(1992), No. 3, 333–367

work page 1992

[6] [6]

T. Ito, R. Tatematsu and T. Tsushima,Criteria of maximality and minimality of van der Geer– van der Vlugt curves, Finite Fields Appl.111(2026), Article ID 102781

work page 2026

[7] [7]

T. Ito, D. Takeuchi and T. Tsushima,TheL-polynomials of van der Geer–van der Vlugt curves in characteristic2, J. Lond. Math. Soc.113(2026), No. 3, Article ID e70523

work page 2026

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Karemaker and R

V. Karemaker and R. Pries,Fully maximal and fully minimal abelian varieties, J. Pure Appl. Algebra223(2019), No. 7, 3031–3056

work page 2019

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Laumon,Semi-continuit´ e du conducteur de Swan (d’apr` es P

G. Laumon,Semi-continuit´ e du conducteur de Swan (d’apr` es P. Deligne), inThe Euler-Poincar´ e characteristic, pp. 173–219, Ast´ erisque, 82–83, Soc. Math. France, Paris

work page

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Stichtenoth,Algebraic Function Fields and Codes, 2nd ed., Graduate Texts in Mathematics, vol

H. Stichtenoth,Algebraic Function Fields and Codes, 2nd ed., Graduate Texts in Mathematics, vol. 254, Springer, 2009. ADDITIVE POLYNOMIAL F ACTORIZATION & V AN DER GEER–V AN DER VLUGT CUR VES 51

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Stichtenoth and C

H. Stichtenoth and C. Xing,On the structure of the divisor class group of a class of curves over finite fields, Arch. Math. (Basel)65(2) (1995), 141–150

work page 1995

[12] [12]

Takeuchi and T

D. Takeuchi and T. Tsushima,Gauss sums and van der Geer–van der Vlugt curves, Bull. Lond. Math. Soc.56(2024), No. 2, 602–623

work page 2024

[13] [13]

Tsushima,Local Galois representations associated to additive polynomials, Manuscr

T. Tsushima,Local Galois representations associated to additive polynomials, Manuscr. Math. 174(2024), No. 3–4, 1151–1182. Department of Mathematics, F aculty of Science, Kyoto University Kyoto, 606– 8502, Japan Email address:tetsushi@math.kyoto-u.ac.jp Department of Mathematics, Institute of Science Tokyo, 2-12-1 Ookayama, Meguro- ku, Tokyo, 152-8551, Ja...

work page 2024