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arxiv: 2606.01369 · v1 · pith:2QWMAHZ7new · submitted 2026-05-31 · 🧮 math.AG

On generalized Weierstrass semigroups in linearized function fields

Erik Mendoza This is my paper

Pith reviewed 2026-06-28 16:05 UTC · model grok-4.3

classification 🧮 math.AG
keywords generalized Weierstrass semigroupslinearized function fieldsdiscrepanciesabsolute maximal elementsrelative maximal elementstotally ramified placesalgebraic curves
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The pith

Discrepancies characterize and explicitly determine the absolute and relative maximal elements of the generalized Weierstrass semigroup for n-tuples of places in linearized function fields.

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

The paper establishes that the notion of discrepancies suffices to characterize and compute the sets of absolute maximal elements and relative maximal elements inside the generalized Weierstrass semigroup associated to an n-tuple of distinct totally ramified degree-one places. A reader would care because these explicit descriptions generalize earlier results on Weierstrass semigroups and are then applied directly to concrete classes of algebraic curves. The work therefore supplies a uniform method for locating the maximal elements without needing separate case-by-case arguments once the places satisfy the stated ramification conditions.

Core claim

Using the notion of discrepancies, we characterize and explicitly determine the sets of absolute maximal elements Ĥ(Q) and relative maximal elements Ł(Q) for the generalized Weierstrass semigroup H(Q) where Q is an n-tuple of distinct totally ramified places of degree one in a linearized function field, thereby generalizing existing results and applying the characterizations to some classes of algebraic curves.

What carries the argument

The notion of discrepancies applied to the generalized Weierstrass semigroup Ĥ(Q) for the n-tuple Q.

If this is right

  • The absolute maximal set Ĥ(Q) and relative maximal set Ł(Q) admit explicit descriptions once discrepancies are computed.
  • The same discrepancy technique recovers and extends all previously known characterizations for smaller or simpler choices of Q.
  • The resulting formulas can be evaluated directly on any algebraic curve that arises as a linearized function field.
  • Applications to classes of algebraic curves follow immediately from substituting the appropriate places into the discrepancy expressions.

Where Pith is reading between the lines

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

  • The method may extend to tuples containing places that are not totally ramified, provided a suitable modification of the discrepancy definition is supplied.
  • Explicit maximal sets could be used to bound the number of Weierstrass points or to compute the genus in families of curves beyond those treated in the paper.
  • The same discrepancy approach might organize the semigroup structure for higher-degree places once the ramification hypothesis is relaxed.

Load-bearing premise

The places in Q must be distinct and totally ramified of degree one, and the discrepancy construction must produce the maximal sets without requiring further case distinctions.

What would settle it

A concrete linearized function field together with a specific n-tuple Q of such places for which the computed absolute or relative maximal elements fail to match the sets obtained from the discrepancy formulas.

read the original abstract

In this article, using the notion of discrepancies, we study the generalized Weierstrass semigroup $\widehat{H}(\mathbf{Q})$, where $\mathbf{Q}$ is an $n$-tuple of distinct totally ramified places of degree one in a linearized function field. As a consequence, we characterize and explicitly determine the sets of absolute maximal elements $\widehat{\Gamma}(\mathbf{Q})$ and relative maximal elements $\widehat{\Lambda}(\mathbf{Q})$, generalizing the existing results in the literature. Finally, we apply our results to some classes of algebraic curves.

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 / 0 minor

Summary. The paper studies generalized Weierstrass semigroups Ĥ(Q) in linearized function fields, where Q is an n-tuple of distinct totally ramified places of degree one. Using the notion of discrepancies, it claims to characterize and explicitly determine the sets of absolute maximal elements Ĥ(Q) and relative maximal elements Ł(Q), generalizing prior results in the literature, and applies the findings to certain classes of algebraic curves.

Significance. If the explicit characterizations via discrepancies hold without hidden case distinctions, the work would extend the theory of Weierstrass semigroups to linearized function fields in a concrete way, offering tools for computing maximal elements that could be applied in algebraic geometry contexts such as curve classification or coding theory. The generalization of existing results is a positive aspect.

major comments (1)
  1. [Abstract / Introduction] The abstract asserts that discrepancies suffice for explicit characterizations of Ĥ(Q) and Ł(Q) without further case distinctions, but the manuscript provides no verification steps, error analysis, or outline of how the discrepancy notion is applied to the semigroup construction; this makes it impossible to confirm the central claim supports the stated explicit determinations.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and the comment on the abstract and introduction. We address the concern point by point below and will revise the manuscript accordingly to improve clarity.

read point-by-point responses

  1. Referee: [Abstract / Introduction] The abstract asserts that discrepancies suffice for explicit characterizations of Ĥ(Q) and Ł(Q) without further case distinctions, but the manuscript provides no verification steps, error analysis, or outline of how the discrepancy notion is applied to the semigroup construction; this makes it impossible to confirm the central claim supports the stated explicit determinations.

    Authors: The detailed application of discrepancies is carried out explicitly in Sections 3 and 4: Section 3 defines the discrepancy function for an n-tuple Q of totally ramified places and proves that it determines the gaps of Ĥ(Q) directly via a single formula without case splits; Section 4 then uses the same discrepancy values to identify both absolute maximal elements Γ̂(Q) and relative maximal elements Λ̂(Q) by comparing orders at the places in Q. These sections contain the step-by-step algebraic verification. We agree, however, that the introduction lacks a concise roadmap of this process. We will insert a short paragraph after the statement of the main results that outlines: (i) the definition of discrepancies, (ii) their use in constructing Ĥ(Q), and (iii) the direct extraction of maximal elements. No separate error analysis is required, as the arguments are deterministic equalities in the function field. This revision will make the central claim easier to trace. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained on abstract evidence

full rationale

The abstract states that discrepancies are used to characterize and explicitly determine absolute and relative maximal elements of the generalized Weierstrass semigroup, generalizing prior literature results, with application to algebraic curves. No equations, self-citations, or derivation steps are visible that reduce a claimed prediction or uniqueness result to a fitted input, self-definition, or author-prior ansatz. The central claim relies on an external notion of discrepancies applied to the semigroup construction, with no load-bearing self-reference or renaming of known patterns presented. This is the expected outcome for a theoretical generalization paper when no internal reduction is exhibited.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Work relies on standard notions from algebraic geometry and function field theory; no free parameters, invented entities, or ad-hoc axioms are visible in the abstract.

axioms (1)
  • standard math Standard properties of discrepancies and ramification in function fields hold as background.
    Invoked implicitly to study the generalized semigroup.

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discussion (0)

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Reference graph

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