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arxiv: 2605.29311 · v1 · pith:SOV4WRZ4new · submitted 2026-05-28 · 🧮 math.NT · math.GR

Weierstrass semigroups at totally ramified places of degree one on linearized function fields

Huachao Zhang , Chang-An Zhao This is my paper

Pith reviewed 2026-06-29 06:07 UTC · model grok-4.3

classification 🧮 math.NT math.GR
keywords Weierstrass semigrouplinearized function fieldtotally ramified placegap setFrobenius numbersymmetric semigroupfunction field
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The pith

For totally ramified degree-one places in linearized function fields, the gap set admits a unified description that determines the Weierstrass semigroup explicitly.

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

The paper shows that in a linearized function field viewed as a Galois extension of a rational function field, the gaps at a totally ramified place of degree one can be described in a unified way. This leads to explicit expressions for the generators of the Weierstrass semigroup, its multiplicity, and its Frobenius number, as well as a condition for the semigroup to be symmetric. The results are then used to find minimal generating sets at multiple such places and to construct examples from separable polynomials.

Core claim

In a linearized function field F over K(x), for a totally ramified place Q of degree one, the set G(Q) of gaps at Q has a unified description. Consequently, the Weierstrass semigroup H(Q) has an explicit system of generators, multiplicity, and Frobenius number, and there is a necessary and sufficient condition for H(Q) to be symmetric. The minimal generating sets at several such places are also described explicitly, along with functions realizing them.

What carries the argument

The unified description of the gap set G(Q) at the totally ramified place Q of degree one

If this is right

  • Explicit formulas for the multiplicity and Frobenius number of H(Q) become available.
  • Symmetry of H(Q) can be checked via a necessary and sufficient condition.
  • Minimal generating sets of H(Q) can be determined for multiple ramified places.
  • Specific functions can be constructed whose pole orders lie in the minimal generating set.

Where Pith is reading between the lines

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

  • The approach may extend to other classes of Galois extensions beyond linearized ones.
  • Applications to algebraic geometry codes could benefit from these explicit semigroups.
  • The symmetry condition might connect to the Riemann-Roch theorem applications in coding theory.

Load-bearing premise

The extension F/K(x) must be a linearized Galois extension and Q must be a totally ramified place of degree one.

What would settle it

Direct computation of the gaps at such a place Q in a specific linearized function field that contradicts the unified description provided.

read the original abstract

A linearized function field $F$ can be viewed as a Galois extension of a rational function field $K(x)$. For a totally ramified place $Q$ of degree one in $F/K(x)$, we give a unified description of the set $G(Q)$ of gaps at $Q$. As a consequence, we explicitly provide a system of generators, the multiplicity, and the Frobenius number of the Weierstrass semigroup $H(Q)$. Moreover, we give a necessary and sufficient condition for $H(Q)$ to be symmetric. Then we investigate the minimal generating set of the Weierstrass semigroups at several totally ramified places of degree one. We not only explicitly describe the minimal generating set, but also provide functions whose coefficients of pole divisors lie in the minimal generating set. Finally, we investigate the linearized function field associated with the denominator of a separable polynomial and apply our results to present several examples.

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

0 major / 3 minor

Summary. The paper studies Weierstrass semigroups H(Q) attached to totally ramified places Q of degree one in linearized Galois extensions F/K(x). It supplies a unified description of the gap set G(Q), from which explicit generators, the multiplicity, the Frobenius number, and a necessary-and-sufficient symmetry criterion for H(Q) are derived. The work then determines minimal generating sets at several such places, exhibits functions realizing those generators via pole divisors, and applies the results to linearized fields arising from denominators of separable polynomials, including concrete examples.

Significance. If the derivations are correct, the explicit, structure-driven formulas for gaps and semigroup invariants in this Galois class constitute a concrete advance over case-by-case computations. The provision of functions whose pole orders lie in the minimal generating set and the symmetry criterion are particularly useful for further applications in algebraic geometry or coding theory over function fields.

minor comments (3)
  1. The abstract and introduction should clarify whether the unified description of G(Q) applies uniformly to all linearized fields or requires additional restrictions on the base field K or the ramification index (see the statement following the definition of linearized function fields).
  2. Notation for the minimal generating set and the pole-divisor functions in the later sections should be introduced with a short table or explicit list to improve readability when multiple places are treated simultaneously.
  3. The examples in the final section would benefit from a brief comparison table listing the computed multiplicity, Frobenius number, and symmetry status for each concrete linearized field considered.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, the recognition of its potential advance in providing unified descriptions and explicit formulas for Weierstrass semigroups in this Galois class, and the recommendation of minor revision. No major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained under stated hypotheses

full rationale

The paper derives an explicit description of the gap set G(Q) and the Weierstrass semigroup H(Q) (including generators, multiplicity, Frobenius number, and symmetry criterion) for totally ramified degree-one places Q in linearized Galois extensions F/K(x). These results are presented as consequences of the Galois action and ramification structure under the linearized property; no step defines a quantity in terms of itself, renames a fitted parameter as a prediction, or relies on a load-bearing self-citation whose content reduces to the present claims. The argument is scoped precisely to the given structural hypotheses without internal reduction to fitted inputs or ansatz smuggling. This is the normal case of an independent derivation within a well-defined class of function fields.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The results rest on the standard theory of algebraic function fields, places, and ramification in Galois extensions; no free parameters or new postulated entities are introduced.

axioms (2)
  • standard math Standard properties of algebraic function fields over finite fields, including the Riemann-Roch theorem, ramification theory in Galois extensions, and the definition of Weierstrass semigroups and gaps.
    Invoked throughout the description of G(Q) and H(Q).
  • domain assumption The function field F is linearized, i.e., a Galois extension of K(x) satisfying the linearized polynomial condition.
    This is the structural hypothesis under which the unified gap description is claimed.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. On generalized Weierstrass semigroups in linearized function fields

    math.AG 2026-05 unverdicted novelty 5.0

    Characterizes absolute and relative maximal elements of generalized Weierstrass semigroups in linearized function fields and applies results to algebraic curves.

Reference graph

Works this paper leans on

40 extracted references · 36 canonical work pages · cited by 1 Pith paper

  1. [2]

    E. A. Mendoza, On Kummer extensions with one place at infinity, Finite Fields and Their Applications 89 (2023) 102209.doi:10.1016/j.ffa.2023.102209

    work page doi:10.1016/j.ffa.2023.102209 2023

  2. [3]

    Abdón, H

    M. Abdón, H. Borges, L. Quoos, Weierstrass points on Kummer extensions, Ad- vances in Geometry 19 (3) (2019) 323–333.doi:10.1515/advgeom-2018-0021

    work page doi:10.1515/advgeom-2018-0021 2019

  3. [4]

    Beelen, L

    P. Beelen, L. Landi, M. Montanucci, Weierstrass semigroups on the Skabelund maximal curve, Finite Fields and Their Applications 72 (2021) 101811.doi:10. 1016/j.ffa.2021.101811

    work page arXiv 2021

  4. [5]

    Bartoli, M

    D. Bartoli, M. Montanucci, G. Zini, Weierstrass semigroups at every point of the Suzuki curve, Acta Arithmetica 197 (1) (2021) 1–20.doi:10.4064/ aa181203-24-2

    2021

  5. [6]

    Online learning: A comprehensive survey

    P. Beelen, M. Montanucci, Weierstrass semigroups on the Giulietti–Korchmáros curve, Finite Fields and Their Applications 52 (2018) 10–29.doi:10.1016/j. ffa.2018.03.002

    work page doi:10.1016/j 2018

  6. [7]

    Cotterill, E

    E. Cotterill, E. A. R. Mendoza, P. Speziali, On gap sets in arbitrary Kummer ex- tensions ofK(x)(2025).arXiv:2506.19169,doi:10.48550/arXiv.2506.19169

    work page doi:10.48550/arxiv.2506.19169 2025

  7. [8]

    Güneri, M

    C. Güneri, M. Özdemiry, H. Stichtenoth, The automorphism group of the gener- alized Giulietti–Korchmáros function field, Advances in Geometry 13 (2) (2013) 369–380.doi:10.1515/advgeom-2012-0040

    work page doi:10.1515/advgeom-2012-0040 2013

  8. [9]

    A. S. Castellanos, A. M. Masuda, L. Quoos, One- and Two-Point Codes Over Kummer Extensions, IEEE Transactions on Information Theory 62 (9) (2016) 4867–4872.doi:10.1109/TIT.2016.2583437

    work page doi:10.1109/tit.2016.2583437 2016

  9. [10]

    Montanucci, V

    M. Montanucci, V. Pallozzi Lavorante, AG codes from the second generalization of the GK maximal curve, Discrete Mathematics 343 (5) (2020) 111810.doi: 10.1016/j.disc.2020.111810

    work page doi:10.1016/j.disc.2020.111810 2020

  10. [11]

    Garcia, S

    A. Garcia, S. J. Kim, R. F. Lax, Consecutive Weierstrass gaps and minimum distance of Goppa codes, Journal of Pure and Applied Algebra 84 (2) (1993) 199– 207.doi:10.1016/0022-4049(93)90039-V. 24

    work page doi:10.1016/0022-4049(93)90039-v 1993

  11. [12]

    Korchmáros, G

    G. Korchmáros, G. Nagy, Hermitian codes from higher degree places, Journal of Pure and Applied Algebra 217 (12) (2013) 2371–2381.doi:10.1016/j.jpaa. 2013.04.002

    work page doi:10.1016/j.jpaa 2013

  12. [13]

    O. Geil, R. Matsumoto, Bounding the number of F q -rational places in algebraic function fields using Weierstrass semigroups, Journal of Pure and Applied Algebra 213 (6) (2009) 1152–1156.doi:10.1016/j.jpaa.2008.11.013

    work page doi:10.1016/j.jpaa.2008.11.013 2009

  13. [14]

    Niemann, Non-isomorphic maximal function fields of genusq−1, Finite Fields and Their Applications 106 (2025) 102618.doi:10.1016/j.ffa.2025.102618

    J. Niemann, Non-isomorphic maximal function fields of genusq−1, Finite Fields and Their Applications 106 (2025) 102618.doi:10.1016/j.ffa.2025.102618

    work page doi:10.1016/j.ffa.2025.102618 2025

  14. [15]

    Beelen, M

    P. Beelen, M. Montanucci, J. Niemann, L. Quoos, A family of non-isomorphic maximal function fields, Mathematische Zeitschrift 309 (2) (2025) 19.doi:10. 1007/s00209-024-03650-1

    2025

  15. [16]

    L. Ma, C. Xing, S. L. Yeo, On automorphism groups of cyclotomic function fields over finite fields, Journal of Number Theory 169 (2016) 406–419.doi:10.1016/ j.jnt.2016.05.026

    2016

  16. [17]

    Montanucci, G

    M. Montanucci, G. Tizziotti, G. Zini, On the automorphism group of a family of maximal curves not covered by the Hermitian curve, Finite Fields and Their Applications 99 (2024) 102498.doi:10.1016/j.ffa.2024.102498

    work page doi:10.1016/j.ffa.2024.102498 2024

  17. [18]

    Beelen, M

    P. Beelen, M. Montanucci, L. Vicino, Weierstrass semigroups and automorphism group of a maximal curve with the third largest genus, Finite Fields and Their Applications 92 (2023) 102300.doi:10.1016/j.ffa.2023.102300

    work page doi:10.1016/j.ffa.2023.102300 2023

  18. [19]

    Beelen, M

    P. Beelen, M. Montanucci, L. Vicino, Weierstrass semigroups and automorphism group of a maximal function field with the third largest possible genus,q≡1 (mod 3), Finite Fields and Their Applications 109 (2026) 102701.doi:10.1016/ j.ffa.2025.102701

    work page arXiv 2026

  19. [20]

    Beelen, M

    P. Beelen, M. Montanucci, L. Vicino, Weierstrass semigroups and automorphism group of a maximal function field with the third largest possible genus,q≡0 (mod 3), Finite Fields and Their Applications 110 (2026) 102729.doi:10.1016/ j.ffa.2025.102729

    work page arXiv 2026

  20. [21]

    267 of Grundlehren Der Mathematischen Wissenschaften, Springer New York, New York, NY, 1985.doi:10.1007/978-1-4757-5323-3

    E.Arbarello, M.Cornalba, P.A.Griffiths, J.Harris, GeometryofAlgebraicCurves, Vol. 267 of Grundlehren Der Mathematischen Wissenschaften, Springer New York, New York, NY, 1985.doi:10.1007/978-1-4757-5323-3

    work page doi:10.1007/978-1-4757-5323-3 1985

  21. [22]

    S. J. Kim, On the index of the Weierstrass semigroup of a pair of points on a curve, Archiv der Mathematik 62 (1) (1994) 73–82.doi:10.1007/BF01200442

    work page doi:10.1007/bf01200442 1994

  22. [23]

    Homma, The Weierstrass semigroup of a pair of points on a curve, Archiv der Mathematik 67 (4) (1996) 337–348.doi:10.1007/BF01197599

    M. Homma, The Weierstrass semigroup of a pair of points on a curve, Archiv der Mathematik 67 (4) (1996) 337–348.doi:10.1007/BF01197599

    work page doi:10.1007/bf01197599 1996

  23. [24]

    Carvalho, F

    C. Carvalho, F. Torres, On Goppa Codes and Weierstrass Gaps at Several Points, Designs, Codes and Cryptography 35 (2) (2005) 211–225.doi:10.1007/ s10623-005-6403-4. 25

    2005

  24. [25]

    G. L. Matthews, The Weierstrass Semigroup of anm-tuple of Collinear Points on a Hermitian Curve, in: G. Goos, J. Hartmanis, J. Van Leeuwen, G. L. Mullen, A. Poli, H. Stichtenoth (Eds.), Finite Fields and Applications, Vol. 2948, Springer Berlin Heidelberg, Berlin, Heidelberg, 2004, pp. 12–24.doi: 10.1007/978-3-540-24633-6_2

    work page doi:10.1007/978-3-540-24633-6_2 2004

  25. [26]

    G. L. Matthews, Weierstrass Semigroups and Codes from a Quotient of the Her- mitian Curve, Designs, Codes and Cryptography 37 (3) (2005) 473–492.doi: 10.1007/s10623-004-4038-5

    work page doi:10.1007/s10623-004-4038-5 2005

  26. [27]

    G. L. Matthews, On Weierstrass Semigroups of Some Triples on Norm-Trace Curves, in: Y. M. Chee, C. Li, S. Ling, H. Wang, C. Xing (Eds.), Coding and Cryptology, Vol. 5557, Springer Berlin Heidelberg, Berlin, Heidelberg, 2009, pp. 146–156.doi:10.1007/978-3-642-01877-0_13

    work page doi:10.1007/978-3-642-01877-0_13 2009

  27. [28]

    G. L. Matthews, J. D. Peachey, Minimal generating sets of Weierstrass semi- groups of certainm-tuples on the norm-trace function field, in: G. McGuire, G. L. Mullen, D. Panario, I. E. Shparlinski (Eds.), Contemporary Mathematics, Vol. 518, American Mathematical Society, Providence, Rhode Island, 2010, pp. 315– 326.doi:10.1090/conm/518/10214

    work page doi:10.1090/conm/518/10214 2010

  28. [29]

    G. L. Matthews, D. Skabelund, M. Wills, Triples of rational points on the Hermi- tian curve and their Weierstrass semigroups, Journal of Pure and Applied Algebra 225 (8) (2021) 106623.doi:10.1016/j.jpaa.2020.106623

    work page doi:10.1016/j.jpaa.2020.106623 2021

  29. [30]

    Tizziotti, A

    G. Tizziotti, A. S. Castellanos, Weierstrass Semigroup and Pure Gaps at Several Points on the GK Curve, Bulletin of the Brazilian Mathematical Society, New Series 49 (2) (2018) 419–429.doi:10.1007/s00574-017-0059-3

    work page doi:10.1007/s00574-017-0059-3 2018

  30. [31]

    Castellanos, G

    A. Castellanos, G. Tizziotti, On Weierstrass semigroup at m points on curves of the formf(y) =g(x), Journal of Pure and Applied Algebra 222 (7) (2018) 1803–1809. doi:10.1016/j.jpaa.2017.08.007

    work page doi:10.1016/j.jpaa.2017.08.007 2018

  31. [32]

    C. Hu, S. Yang, Multi-point codes from the GGS curves, Advances in Mathematics of Communications 14 (2) (2020) 279–299.doi:10.3934/amc.2020020

    work page doi:10.3934/amc.2020020 2020

  32. [33]

    A. S. Castellanos, M. Bras-Amorós, Weierstrass semigroup atm+1rational points in maximal curves which cannot be covered by the Hermitian curve, Designs, Codes and Cryptography 88 (8) (2020) 1595–1616.doi:10.1007/s10623-020-00757-4

    work page doi:10.1007/s10623-020-00757-4 2020

  33. [34]

    S. Yang, C. Hu, Weierstrass semigroups from Kummer extensions, Finite Fields and Their Applications 45 (2017) 264–284.doi:10.1016/j.ffa.2016.12.005

    work page doi:10.1016/j.ffa.2016.12.005 2017

  34. [35]

    C. Hu, S. Yang, Multi-point codes over Kummer extensions, Designs, Codes and Cryptography 86 (1) (2018) 211–230.doi:10.1007/s10623-017-0335-7

    work page doi:10.1007/s10623-017-0335-7 2018

  35. [36]

    A.S.Castellanos, E.Mendoza, G.Tizziotti, OngeneralizedWeierstrasssemigroups in arbitrary Kummer extensions ofFq(x), Finite Fields and Their Applications 112 (2026) 102808.doi:10.1016/j.ffa.2026.102808. 26

    work page doi:10.1016/j.ffa.2026.102808 2026

  36. [37]

    S. Yang, C. Hu, Weierstrass semigroups on the third function field in a tower at- taining the Drinfeld-Vlăduţ bound, Advances in Mathematics of Communications 18 (4) (2024) 1051–1083.doi:10.3934/amc.2022066

    work page doi:10.3934/amc.2022066 2024

  37. [38]

    H. Navarro, Bases for Riemann–Roch spaces of linearized function fields with appli- cations to generalized algebraic geometry codes, Designs, Codes and Cryptography 92 (10) (2024) 3033–3048.doi:10.1007/s10623-024-01426-6

    work page doi:10.1007/s10623-024-01426-6 2024

  38. [39]

    Maharaj, Code Construction on Fiber Products of Kummer Covers, IEEE Transactions on Information Theory 50 (9) (2004) 2169–2173.doi:10.1109/TIT

    H. Maharaj, Code Construction on Fiber Products of Kummer Covers, IEEE Transactions on Information Theory 50 (9) (2004) 2169–2173.doi:10.1109/TIT. 2004.833356

    work page doi:10.1109/tit 2004

  39. [40]

    A. S. Castellanos, E. A. R. Mendoza, L. Quoos, Weierstrass semigroups, pure gaps and codes on function fields, Designs, Codes and Cryptography 92 (5) (2024) 1219–1242.doi:10.1007/s10623-023-01339-w

    work page doi:10.1007/s10623-023-01339-w 2024

  40. [41]

    Stichtenoth, Algebraic Function Fields and Codes, 2nd Edition, no

    H. Stichtenoth, Algebraic Function Fields and Codes, 2nd Edition, no. 254 in Graduate Texts in Mathematics, Springer Berlin Heidelberg, Berlin, 2009.doi: 10.1007/978-3-540-76878-4. 27

    work page doi:10.1007/978-3-540-76878-4 2009