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Paper IX · Lean source

InvalidPrescribedRectangularUniqueness.lean

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polynomial_regularity/controls/InvalidPrescribedRectangularUniqueness.lean18 lines

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1 module
2 public import PolynomialParameterTowerControls
3 @[expose] public section
4 open Komorebi.PolynomialParameterTowerControls Komorebi.PolynomialPadicLift
5 open Komorebi.PolynomialParameterSystem
6 
7 -- Invalid: even solved-block surjectivity does not force equal-parameter uniqueness
8 -- when there are more solved variables than equations. The positive control gives
9 -- explicit distinct roots in the same initial ball with exactly the same parameter.
10 example : Function.Surjective
11     (Komorebi.PolynomialPrescribedSystem.solvedJacobian 2 firstSolved (fun _ => 0)) →
12     ∀ b c : Fin (2 + 1) → ℤ_[2],
13       (∀ i, padicEval 2 firstSolved b i = 0) →
14       (∀ i, padicEval 2 firstSolved c i = 0) →
15       (∀ i, (2 : ℤ_[2])^1 ∣ b i - ((0 : Fin (2 + 1) → ℤ) i : ℤ_[2])) →
16       (∀ i, (2 : ℤ_[2])^1 ∣ c i - ((0 : Fin (2 + 1) → ℤ) i : ℤ_[2])) →
17       freePart (e := 2) (q := 1) b = freePart c → b = c := by
18   exact rectangular_same_parameter_nonunique

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Back to Paper IX: Arithmetic Packets, Dyadic Towers, and Local Polynomial Deformation