OAI.QuarticPowerFree.medium_degree_dyadic_tail: intermediate lemma from OpenAI, Squarefree values of quartics and power-free values of polynomials

Proved

OAI.QuarticPowerFree.medium_degree_dyadic_tail

by dbenbenn · Oct 25, 2026 · Mathlib 0df444a (Lean v4.33.1)

openai-math

An intermediate lemma of the split of OpenAI's proof of OAI.QuarticPowerFree.allDegrees, carved from OpenAI's openai/math development for this platform. The statement and its proof are OpenAI's (OpenAI, Squarefree values of quartics and power-free values of polynomials, OpenAI Math Release, September 24, 2026, result family 20, github.com/openai/math, Apache License 2.0); in Lean it is the declaration OAI.QuarticPowerFree.medium_degree_dyadic_tail of lean/OAI/NumberTheory/PowerFree/MediumDegree.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.QuarticPowerFree, with open Filter NumberField, open scoped Topology Classical):

theorem medium_degree_dyadic_tail (f : Polynomial ℤ) (hf : f.IsPrimitive)
    (hirr : Irreducible (f.map (Int.castRingHom ℚ))) (hc : 0 < f.leadingCoeff)
    (hd : 5 ≤ f.natDegree) (hd' : f.natDegree ≤ 8) :
    ∃ C : ℝ, 0 < C ∧ ∀ᶠ N : ℕ in atTop,
      (dyadicTailCount f (f.natDegree-2) N:ℝ) ≤ C*(N:ℝ)^(1-3/1000:ℝ)

Definitions. The statement is written with OAI.QuarticPowerFree.dyadicTailCount (definitions bundle OAIB3Qp000 of this split).

Used by. The proof of OAI.QuarticPowerFree.allDegrees of this split imports it.

Preamble

import Mathlib
import Definitions.Def_OAIB3Qp000

Formal statement

section

set_option linter.all false
set_option linter.unusedVariables false
set_option linter.unusedTactic false
set_option linter.unreachableTactic false

namespace OAI

namespace QuarticPowerFree

section
open Filter NumberField
open scoped Topology Classical

theorem medium_degree_dyadic_tail (f : Polynomial ℤ) (hf : f.IsPrimitive)
    (hirr : Irreducible (f.map (Int.castRingHom ℚ))) (hc : 0 < f.leadingCoeff)
    (hd : 5 ≤ f.natDegree) (hd' : f.natDegree ≤ 8) :
    ∃ C : ℝ, 0 < C ∧ ∀ᶠ N : ℕ in atTop,
      (dyadicTailCount f (f.natDegree-2) N:ℝ) ≤ C*(N:ℝ)^(1-3/1000:ℝ) := by
  sorry

end

open MvPolynomial
open scoped Classical BigOperators

attribute [instance] QuarticBoxSystem.field QuarticBoxSystem.charZero

end QuarticPowerFree

end OAI
end

Source

OpenAI, Squarefree values of quartics and power-free values of polynomials, OpenAI Math Release, September 24, 2026, https://github.com/openai/math (result family 20; paper: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Squarefree-values-of-quartics-and-power-free-values-of-polynomials-September-24-2026/manuscript.pdf; Apache License 2.0); Lean: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/NumberTheory/PowerFree/MediumDegree.lean (declaration OAI.QuarticPowerFree.medium_degree_dyadic_tail)

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