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arxiv: 2606.10584 · v1 · pith:VCDCHEHZnew · submitted 2026-06-09 · 🧮 math.NT · math.AG

Hilbert irreducibility for algebraic points

Borys Kadets , Danny Neftin This is my paper

Pith reviewed 2026-06-27 11:51 UTC · model grok-4.3

classification 🧮 math.NT math.AG
keywords Hilbert irreducibilityreducible fiberscurve coveringsalgebraic pointsindecomposable mapsbranch pointsnumber fields
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The pith

For fixed curves, high-degree indecomposable maps to the line have finitely many reducible fibers over points of degree below b/7-2 after exclusions.

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

The paper studies coverings phi from X to X0 over a number field and asks for which d the degree-d points p on X0 with reducible fiber phi inverse of p over k(p) form a finite set. It proves multiple theorems that guarantee finiteness in this setting. When X has infinitely many degree-d points the images of those points are first removed from consideration. The main consequence is that any fixed curve X admits only finitely many such reducible fibers for all sufficiently high-degree indecomposable rational functions phi from X to the projective line that have b branch points, provided d is less than b over 7 minus 2.

Core claim

Given a covering phi colon X to X0 over a number field k and an integer d, the set of degree-d points p in X0 of k-bar such that the fiber phi inverse of p is reducible over k(p) is finite; when X itself has infinitely many degree-d points their images are removed first. As a consequence, for any fixed curve X and all sufficiently high-degree indecomposable rational functions phi colon X to P1 with b branch points, the set of reducible fibers above degree-d points with d less than b over 7 minus 2, not containing a degree-d point from X, is finite.

What carries the argument

Indecomposable rational functions phi colon X to P1 of high degree together with the linear bound d less than b over 7 minus 2 on the degree of base points, which together force finiteness of the reducible fibers.

Load-bearing premise

The maps must be indecomposable and of sufficiently high degree, with the degree d required to satisfy the explicit inequality d less than b over 7 minus 2.

What would settle it

An explicit fixed curve X together with an indecomposable high-degree phi having b branch points, plus an infinite sequence of degree-d points p with d less than b over 7 minus 2 such that every fiber is reducible over k(p) yet contains no degree-d point of X.

read the original abstract

We study the following problem: given a covering of curves $\phi\colon X \to X_0$ over a number field $k$, and an integer $d$, when is the set \[\{p \in X_0(\overline{k})|\ \mathrm{deg}\ p = d, \text{ and the fiber } \phi^{-1}(p) \text{ is reducible over } k(p)\}\] finite? In case $X$ itself admits infinitely many degree $d$ points, we consider the modified problem where the images of degree $d$ points on $X$ are removed from the set. We prove a number of theorems ensuring a positive answer. As a consequence we show that for a fixed curve $X$ and all sufficiently high-degree indecomposable rational functions $\phi:X \to \mathbb{P}^1$ with $b$ branch points, the set of reducible fibers above degree $d<b/7-2$ points, not containing a degree $d$ point from $X$, is finite.

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 / 2 minor

Summary. The paper studies finiteness questions for reducible fibers of coverings φ: X → X0 over number fields, specifically the set of degree-d points p on X0 where φ^{-1}(p) is reducible over k(p). When X has infinitely many degree-d points, the images of those points are excluded. Several theorems are proved guaranteeing finiteness under suitable hypotheses on φ. As a consequence, for fixed X and all sufficiently high-degree indecomposable rational functions φ: X → ℙ¹ with b branch points, the set of reducible fibers over degree-d points with d < b/7 − 2 (excluding images of degree-d points on X) is finite.

Significance. If the main theorems hold, the work supplies new explicit Hilbert-irreducibility statements for algebraic points of bounded degree, with a concrete numerical bound in the rational-function case. The indecomposability and high-degree hypotheses are standard in the area; the explicit bound d < b/7 − 2 is a concrete, falsifiable output that could be tested on low-genus examples.

major comments (1)
  1. [Abstract / Consequence statement] The abstract states that the bound d < b/7 − 2 arises as a consequence of the main theorems, yet the derivation of the numerical coefficient 1/7 is not visible from the given statement. If this coefficient is obtained by optimizing constants in an earlier theorem (e.g., via Riemann–Hurwitz or monodromy arguments), the optimization step should be recorded explicitly so that the bound can be verified or improved.
minor comments (2)
  1. [Abstract] The notation “X0” for the base curve is introduced without an explicit definition in the abstract; a sentence clarifying that X0 is the target of the covering φ would improve readability.
  2. [Abstract] The phrase “all sufficiently high-degree” is used for the indecomposable maps φ; a quantitative lower bound on deg(φ) in terms of genus(X) or b would make the statement fully effective.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and positive recommendation. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract / Consequence statement] The abstract states that the bound d < b/7 − 2 arises as a consequence of the main theorems, yet the derivation of the numerical coefficient 1/7 is not visible from the given statement. If this coefficient is obtained by optimizing constants in an earlier theorem (e.g., via Riemann–Hurwitz or monodromy arguments), the optimization step should be recorded explicitly so that the bound can be verified or improved.

    Authors: We agree that the origin of the coefficient 1/7 should be made explicit. It arises from optimizing the constants in the main finiteness theorems (via the Riemann–Hurwitz formula applied to the Galois closure together with the indecomposability hypothesis and the lower bound on the number of branch points). In the revised version we will add a short paragraph immediately after the consequence statement that records the optimization steps, allowing the bound to be verified or sharpened. revision: yes

Circularity Check

0 steps flagged

No circularity: result derived from external theorems in algebraic geometry and number theory

full rationale

The paper establishes finiteness results for sets of points with reducible fibers under explicit hypotheses (indecomposability, sufficiently high degree, and the bound d < b/7-2). These hypotheses are stated as inputs rather than derived outputs, and the consequence is presented as following from proved theorems that invoke standard tools from the field. No self-definitional steps, fitted parameters renamed as predictions, or load-bearing self-citations reducing the central claim to prior author work are detectable in the abstract or stated claims. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The setup relies on standard background from algebraic geometry and number theory; no free parameters, invented entities, or ad-hoc axioms are visible in the abstract.

axioms (2)
  • standard math Standard properties of algebraic curves, their morphisms, and fields of definition over number fields k
    The problem is phrased in terms of coverings of curves over k and degrees of points in X0(k-bar).
  • domain assumption Existence of indecomposable rational functions of high degree with a given number b of branch points
    The consequence assumes such functions exist for any fixed X.

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Reference graph

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