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arxiv: 2605.21513 · v1 · pith:QRW3LSLLnew · submitted 2026-05-15 · 🧮 math.NT · math.CO

On the Natural Density of Monic Integer Polynomials with Roots in a Fixed Number Field

Amirali Fatehizadeh This is my paper

Pith reviewed 2026-05-22 01:17 UTC · model grok-4.3

classification 🧮 math.NT math.CO
keywords natural densitymonic polynomialsnumber fieldsasymptotic estimatesMahler measureDedekind zeta functiongeometry of numbersreducible polynomials
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The pith

The natural density of monic integer polynomials of degree n with a root in fixed number field K vanishes at a rate depending on n, with O(H^{-1} log H) for n=2 and O(H^{-1}) otherwise.

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

This paper examines the family of monic integer polynomials of degree n that have at least one root in a fixed number field K. Although the family is thin and has natural density zero among all monic polynomials of bounded height, the work quantifies the rate at which the density approaches zero. It establishes a phase transition in this rate driven by the degrees of the irreducible factors: quadratic polynomials decay more slowly while higher-degree cases are dominated by those with rational roots. A hybrid method combines the Mahler measure, Dirichlet's unit theorem, and residue analysis of the Dedekind zeta function to derive the explicit bounds. Geometry of numbers supplies additional combinatorial estimates for the reducible and irreducible parts.

Core claim

The central claim is that the natural density of monic integer polynomials of degree n having at least one root in a fixed number field K tends to zero, with the rate of convergence depending on n through a phase transition induced by the degrees of the factors. For n=2 the bound is O(H^{-1} log H), while for n greater than 2 the contribution of rational roots yields O(H^{-1}). This is achieved via a hybrid approach combining the Mahler measure, Dirichlet's unit theorem, and residue analysis of the Dedekind zeta function, supplemented by geometry of numbers to count reducible and irreducible components explicitly.

What carries the argument

The phase transition in asymptotic decay rate induced by the degrees of the polynomial factors, quantified by a hybrid analytic approach that integrates the Mahler measure, Dirichlet's unit theorem, and residues of the Dedekind zeta function.

If this is right

  • For degree 2 the count includes an extra logarithmic factor arising from the contribution of units in the number field.
  • For degrees above 2 the main term is produced by polynomials possessing a rational root, which simplifies the leading asymptotic.
  • The geometry-of-numbers bounds supply explicit constants for the number of reducible polynomials that can be used in direct computation.
  • The estimates hold uniformly for any fixed number field K without further conditions on its discriminant or unit rank.

Where Pith is reading between the lines

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

  • Numerical enumeration of polynomials for moderate H and small n could directly verify whether the predicted phase transition appears in practice.
  • The same hybrid method might be applied to count polynomials with roots in a finite union of number fields, potentially revealing further transitions.
  • The explicit reducible-component bounds could be compared against existing counts from Hilbert irreducibility to test consistency across different thin-set problems.

Load-bearing premise

The hybrid approach integrating the Mahler measure, Dirichlet's unit theorem, and residue analysis of the Dedekind zeta function correctly produces the claimed phase transition and explicit bounds without additional restrictions on the number field K or the degree n beyond those stated.

What would settle it

A direct count for a concrete number field K and degree n=2 showing that the number of qualifying polynomials of height at most H exceeds any constant multiple of H^{-1} log H for infinitely many H would falsify the stated upper bound.

read the original abstract

In this article, we investigate the statistical distribution and asymptotic behavior of the family of monic integer polynomials of degree $n$ having at least one root in a fixed number field $K$. Although the framework of thin sets implies that the natural density of this family in the parameter space of bounded height is zero, explicitly quantifying this vanishing rate is a central challenge in arithmetic statistics. Employing a hybrid approach that integrates the Mahler measure, Dirichlet's unit theorem, and residue analysis of the Dedekind zeta function, we demonstrate that the rate of convergence of this density to zero is strictly dependent on the degree $n$. Specifically, we prove that the degrees of the factors induce a phase transition in the asymptotic behavior; for polynomials of degree $n = 2$, the decay rate is bounded by $O(H^{-1} \log H)$, whereas for higher degrees, the asymptotic behavior is dominated by the contribution of rational roots, yielding a bound of $O(H^{-1})$. Beyond deriving these asymptotic estimates, we apply principles from the geometry of numbers to establish explicit combinatorial bounds for counting both the reducible and irreducible components of these polynomials. These explicit bounds provide practical tools for computational evaluations within this domain.

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 paper investigates the natural density of monic integer polynomials of fixed degree n that have at least one root in a given number field K. It claims that this density vanishes as the height H tends to infinity, with a phase transition in the rate: O(H^{-1} log H) when n=2 arising from a hybrid Mahler-measure/Dirichlet-unit/zeta-residue analysis, and O(H^{-1}) when n>2 because the count is dominated by polynomials with a rational root. The manuscript also supplies explicit combinatorial bounds, obtained via the geometry of numbers, on the reducible and irreducible components of such polynomials.

Significance. If the claimed phase transition and the domination argument are rigorously established, the result would give a concrete quantitative refinement of the thin-set phenomenon in the space of monic polynomials, distinguishing the quadratic case from higher degrees. The hybrid analytic-geometric method is potentially useful for similar counting problems in arithmetic statistics.

major comments (2)
  1. [asymptotic analysis for n>2] The central claim that, for n>2, the contribution of non-rational roots in K is o(H^{-1}) relative to the rational-root contribution is load-bearing for the asserted O(H^{-1}) bound and the phase transition. The passage from Dedekind-zeta residues to polynomial counts via the Mahler measure appears to rely on uniformity of the regulator and class-number estimates that may grow with n; without an explicit comparison of error terms (e.g., in the section deriving the n>2 asymptotic), it is unclear whether the domination holds.
  2. [hybrid method section] The hybrid argument for the n=2 bound O(H^{-1} log H) invokes the residue of the Dedekind zeta function together with the volume of the unit lattice, yet the manuscript supplies no derivation steps or explicit error-term control showing that these combine without n-dependent gaps that would erase the phase transition.
minor comments (2)
  1. [abstract] The abstract states that 'explicit combinatorial bounds' are obtained but does not indicate in which section or theorem these bounds appear, nor does it record the dependence on the degree n or the field K.
  2. [introduction] Notation for the height H and for the parameter space of monic polynomials should be fixed at the beginning of §1 to avoid ambiguity when comparing the n=2 and n>2 regimes.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments, which help clarify the presentation of the asymptotic results. We address each major comment below and will incorporate the suggested clarifications in the revised version.

read point-by-point responses

  1. Referee: [asymptotic analysis for n>2] The central claim that, for n>2, the contribution of non-rational roots in K is o(H^{-1}) relative to the rational-root contribution is load-bearing for the asserted O(H^{-1}) bound and the phase transition. The passage from Dedekind-zeta residues to polynomial counts via the Mahler measure appears to rely on uniformity of the regulator and class-number estimates that may grow with n; without an explicit comparison of error terms (e.g., in the section deriving the n>2 asymptotic), it is unclear whether the domination holds.

    Authors: We appreciate the referee drawing attention to the need for explicit error-term comparisons to confirm the domination argument for n>2. In the manuscript, the leading contribution for n>2 arises from polynomials with a rational root, whose count is asymptotically c_K H^{-1} (with c_K >0 depending only on the fixed field K) via the residue of the Dedekind zeta function at s=1. The contribution from roots in K that are not rational is controlled by applying the geometry-of-numbers bounds on the number of irreducible factors of degree at least 2; because K is fixed, the relevant regulator, class number, and unit-lattice volume are absolute constants independent of n. We will add a new subsection that explicitly compares the main term with the error terms arising from the Mahler-measure estimates and the geometry-of-numbers volume computations, thereby verifying that the non-rational contribution is indeed o(H^{-1}). revision: yes

  2. Referee: [hybrid method section] The hybrid argument for the n=2 bound O(H^{-1} log H) invokes the residue of the Dedekind zeta function together with the volume of the unit lattice, yet the manuscript supplies no derivation steps or explicit error-term control showing that these combine without n-dependent gaps that would erase the phase transition.

    Authors: We thank the referee for noting the absence of detailed derivation steps in the hybrid analysis for the n=2 case. The argument combines the residue of zeta_K(s) at s=1 (which encodes the class number and regulator of the fixed field K) with the covolume of the unit lattice furnished by Dirichlet's unit theorem, then integrates these against the region defined by the Mahler measure of the polynomial being at most H. Since K is fixed, all arithmetic invariants are constants and no n-dependent gaps appear; the logarithmic factor for n=2 originates from the two-dimensional unit lattice, while for n>2 the rational-root term dominates and produces the stricter O(H^{-1}) bound. In the revision we will expand the hybrid-method section to include the complete chain of estimates, explicit error bounds, and a direct comparison that preserves the phase transition between n=2 and n>2. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external standard results

full rationale

The paper's hybrid approach invokes the Mahler measure, Dirichlet's unit theorem, and Dedekind zeta function residues as external inputs. These are standard, independently established results in algebraic number theory with no indication from the abstract or reader's summary that they reduce to self-citations, fitted parameters renamed as predictions, or definitions internal to this work. The claimed phase transition and bounds for different n arise from comparing contributions (rational roots vs. others), but absent any quoted equation showing the output equals the input by construction, the central claims remain independent of the paper's own fitted quantities. This is the common case of a self-contained derivation against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard theorems of algebraic number theory; no free parameters, new entities, or ad-hoc axioms are introduced in the abstract.

axioms (2)
  • domain assumption The framework of thin sets implies that the natural density of this family in the parameter space of bounded height is zero
    Invoked as the starting point for quantifying the vanishing rate.
  • standard math Dirichlet's unit theorem and residue analysis of the Dedekind zeta function apply directly to the fixed number field K
    Used as core ingredients in the hybrid approach.

pith-pipeline@v0.9.0 · 5743 in / 1349 out tokens · 56558 ms · 2026-05-22T01:17:53.616861+00:00 · methodology

discussion (0)

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Reference graph

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