Finite N3 root coefficient norm le ratio

Proved

D1Bound5.finiteN3_root_coefficient_norm_le_ratio

by tp · Oct 24, 2026 · Mathlib 0df444a (Lean v4.33.1)

diophantine-tuplesnumber-theory

Auxiliary result finiteN3_root_coefficient_norm_le_ratio in the proof of the bound five for Diophantine D(1) sets in imaginary quadratic integer rings. The formal statement specifies its hypotheses and conclusion; a mathematical prose description remains to be supplied.

Preamble

import Lean.Elab.Tactic.Omega
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Normed.Ring.Basic
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.Tactic.Convert
import Mathlib.Tactic.FinCases
import Mathlib.Tactic.GCongr
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.Positivity
import Mathlib.Tactic.Ring
import Definitions.Def_D1Bound5_FiniteN3RatioHeight_Definitions

Formal statement

section
noncomputable section

namespace D1Bound5











theorem _root_.D1Bound5.finiteN3_root_coefficient_norm_le_ratio (x y : ℂ) (r : ℝ)
    (hr : 0 ≤ r) (hxy : ‖x‖ ≤ r*‖y‖) (i : Fin 3) (ell : Fin 4) :
    ‖finiteN3RatioRootCoeff x y i 0 ell‖+‖finiteN3RatioRootCoeff x y i 1 ell‖ ≤
      finiteN3RatioWeights r i ell*‖y‖^(4+(ell : ℕ)) := by sorry





end D1Bound5
end

Source

D1Bound5/FiniteN3RatioHeight.lean, lines 42-62, SHA-256 3b19adaab7a046dd5d462fa93e9c3522cc668117966f660bde17388fd5b6b587. Project: https://github.com/tomislav-math/d1-bound5-lean

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