Finite N2 root coefficient norm le ratio

Proved

D1Bound5.finiteN2_root_coefficient_norm_le_ratio

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

diophantine-tuplesnumber-theory

Auxiliary result finiteN2_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_FiniteN2RatioHeight_Definitions

Formal statement

section
noncomputable section

namespace D1Bound5











theorem _root_.D1Bound5.finiteN2_root_coefficient_norm_le_ratio (x y : ℂ) (r : ℝ)
    (hr : 0 ≤ r) (hxy : ‖x‖ ≤ r*‖y‖) (i : Fin 3) (ell : Fin 3) :
    ‖finiteN2RatioRootCoeff x y i 0 ell‖+‖finiteN2RatioRootCoeff x y i 1 ell‖ ≤
      finiteN2RatioWeights r i ell*‖y‖^(2+(ell : ℕ)) := by sorry





end D1Bound5
end

Source

D1Bound5/FiniteN2RatioHeight.lean, lines 38-54, SHA-256 acd86dd6afa56a2834a76abb67f2f38aa493cf97b505b550e022ae5087305669. Project: https://github.com/tomislav-math/d1-bound5-lean

View graph