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arxiv: 2411.01832 · v3 · pith:UOLJWWFJnew · submitted 2024-11-04 · 🧮 math.AG · math.NT

Construction of Curves with a Controlled First Slope using p-Symmetric Numbers

Robert Moore , Hui June Zhu This is my paper

Pith reviewed 2026-05-23 18:00 UTC · model grok-4.3

classification 🧮 math.AG math.NT
keywords Artin-Schreier curvesfirst slopep-adic weightp-symmetryNewton polygonalgebraic curvesfinite fieldszeta functions
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The pith

Artin-Schreier curves achieve first slope 1/s_p(v) exactly when their unique highest p-adic weight term is p-symmetric.

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

The paper links the first Newton slope of the Artin-Schreier curve y^p - y = f(x) to the p-adic weights of the monomials appearing in f. When one term v has strictly maximal weight, the slope reaches the lower bound 1/s_p(v) if and only if v obeys a combinatorial condition the authors call p-symmetry. Using this equivalence they produce explicit polynomials that realize any desired slope 1/n for n greater than 2, in every characteristic p. A reader cares because the first slope controls the leading term of the zeta function and therefore the distribution of points over finite fields.

Core claim

If the maximal p-adic weight element v in Supp(f) is unique, the first slope of X_f equals 1/s_p(v) if and only if v is p-symmetric. The authors define p-symmetry as the required combinatorial p-adic condition on v and apply it to construct, for every prime p and every integer n > 2, an explicit f whose curve has first slope exactly 1/n.

What carries the argument

p-symmetry, the combinatorial p-adic condition on an exponent v that forces the Newton polygon of the Artin-Schreier equation to attain the slope lower bound 1/s_p(v).

If this is right

  • Explicit families of curves exist in every characteristic with first slope exactly 1/n for any fixed n>2.
  • The first slope is completely determined by the p-adic weight and the symmetry status of the single heaviest term.
  • The construction is combinatorial and therefore works uniformly across all primes p.
  • p-symmetry supplies a sufficient and necessary criterion that converts a number-theoretic condition into a geometric slope.

Where Pith is reading between the lines

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

  • The same symmetry condition may control higher slopes once the maximal term is fixed.
  • p-symmetric exponents could be used to engineer Newton polygons with prescribed initial segments in other classes of covers.
  • Computational searches for small p and small degrees could enumerate all p-symmetric v up to a given size and list the corresponding minimal-slope curves.
  • Relaxing uniqueness of the maximal weight might still allow slope control if the several leading terms satisfy compatible symmetry relations.

Load-bearing premise

The p-adic weight of each exponent in f is defined in the usual way for Artin-Schreier covers, and every p-symmetric v can be realized by an actual polynomial over a field of characteristic p without further hidden constraints.

What would settle it

Exhibit an explicit polynomial f whose unique maximal-weight exponent v is p-symmetric yet the first slope of the resulting curve is strictly larger than 1/s_p(v), or a non-p-symmetric v whose curve nevertheless meets the bound.

read the original abstract

This paper establishes a constructive link between the first slope of Artin-Schreier curves X_f: y^p-y=f(x) and the p-adic weight of the support of f(x). If the maximal p-adic weight element v in Supp(f) is unique, we show that the first slope's lower bound of 1/s_p(v) is achieved if and only if v satisfies a combinatorial p-adic condition, which we define as p-symmetry. As an application, we construct explicit families of curves in every characteristic p with first slope equal to 1/n for every n>2.

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

2 major / 2 minor

Summary. The manuscript establishes an if-and-only-if criterion for Artin-Schreier curves X_f : y^p - y = f(x): if the maximal p-adic weight element v in Supp(f) is unique, then the first slope equals the lower bound 1/s_p(v) precisely when v satisfies the combinatorial condition of p-symmetry. As an application, it constructs explicit families of polynomials realizing first slope 1/n for every n>2 in every characteristic p.

Significance. If the central claim holds, the work supplies a combinatorial criterion for achieving the minimal first slope in Artin-Schreier covers together with parameter-free explicit constructions for every rational slope of the form 1/n. These features directly support the study of Newton polygons of curves in positive characteristic by furnishing concrete, realizable examples without fitted parameters.

major comments (2)
  1. [Application section] The uniqueness hypothesis on the maximal-weight element v is load-bearing for the if-and-only-if statement; the constructions in the application section must be shown to satisfy this hypothesis for the claimed slopes 1/n.
  2. [Definitions preceding the main theorem] The definition of the p-adic weight function and of s_p(v) must be stated explicitly and shown to coincide with the standard notions used for the Newton polygon of Artin-Schreier covers; any deviation would affect the lower-bound claim.
minor comments (2)
  1. The title refers to 'p-Symmetric Numbers' while the text uses 'p-symmetry'; adopt a single term throughout.
  2. An illustrative computation of s_p(v) and the p-symmetry check for a small explicit polynomial would improve readability of the combinatorial condition.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments. We address each of the major comments below.

read point-by-point responses

  1. Referee: [Application section] The uniqueness hypothesis on the maximal-weight element v is load-bearing for the if-and-only-if statement; the constructions in the application section must be shown to satisfy this hypothesis for the claimed slopes 1/n.

    Authors: We agree with this observation. The uniqueness of the maximal p-adic weight element is indeed essential for the if-and-only-if criterion. In the explicit families constructed in the application section to achieve first slope 1/n, the support of f is designed such that there is a unique maximal-weight element. We will add a short paragraph in the revised version explicitly confirming that this hypothesis holds for each of the constructed families. revision: yes

  2. Referee: [Definitions preceding the main theorem] The definition of the p-adic weight function and of s_p(v) must be stated explicitly and shown to coincide with the standard notions used for the Newton polygon of Artin-Schreier covers; any deviation would affect the lower-bound claim.

    Authors: We will revise the manuscript to state the definitions of the p-adic weight function and s_p(v) more explicitly at the beginning of the relevant section. Additionally, we will include a remark or short subsection demonstrating that these definitions coincide with the standard ones appearing in the literature on the Newton polygons of Artin-Schreier covers, thereby confirming that the lower bound 1/s_p(v) is the expected one. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained from definitions

full rationale

The central result is an if-and-only-if equivalence: under uniqueness of maximal p-adic weight v in Supp(f), the first slope equals 1/s_p(v) precisely when v meets the newly defined combinatorial condition of p-symmetry. The p-adic weight and support are standard for Artin-Schreier covers; p-symmetry is introduced as a fresh combinatorial predicate on p-adic digits. Explicit polynomial constructions realizing slope 1/n are asserted without reference to fitted parameters or prior self-citations that would reduce the claim. No load-bearing step collapses to a self-definition, fitted input renamed as prediction, or imported uniqueness theorem. The derivation therefore rests on independent combinatorial verification rather than circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The paper rests on standard facts about Artin-Schreier covers and p-adic weights; it introduces one new combinatorial predicate (p-symmetry) whose independent evidence is the equivalence proof itself.

axioms (1)
  • domain assumption Standard definitions and properties of Artin-Schreier curves, p-adic weights on monomials, and the first slope of their Newton polygons.
    Invoked throughout the link between support of f and the slope.
invented entities (1)
  • p-symmetry no independent evidence
    purpose: Combinatorial condition on the maximal-weight term that is equivalent to achieving the slope lower bound.
    Newly defined in the paper; no external evidence supplied beyond the claimed equivalence.

pith-pipeline@v0.9.0 · 5622 in / 1334 out tokens · 56133 ms · 2026-05-23T18:00:51.484592+00:00 · methodology

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Reference graph

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