JacobianConjecture.lean`

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

11 declarations, 1 contain sorry, 0 declared as axioms

Jacobian conjecture

Reference: Wikipedia

Imports (1)

FormalConjecturesUtil

Imported by 1 file

FormalConjectures/Arxiv/math.0608009/PoissonConjecture.lean

abbrev JacobianConjecture.F Alpöge/Fable's counterexample: a polynomial self-map of k³ with Jacobian determinant -2 which is not injective line 41 source
Source documentation
Alpöge/Fable's counterexample: a polynomial self-map of k³ with Jacobian determinant -2 which is not injective.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, MvPolynomial.RegularFunction, Fin, OfNat.ofNat, Nat
Proof uses: Matrix.vecCons, MvPolynomial, HAdd.hAdd, Finsupp, HMul.hMul, HPow.hPow, MvPolynomial.X, HSub.hSub, Matrix.vecEmpty
38 instances, coercions and auxiliary constants left out.
```
noncomputable abbrev F [CommRing k] : RegularFunction k (Fin 3) (Fin 3) :=
…
```
abbrev JacobianConjecture.G A variant of Alpöge/Fable's counterexample: a polynomial self-map of k³ with Jacobian determinant 1 which is not injective line 48 source
Source documentation
A variant of Alpöge/Fable's counterexample: a polynomial self-map of k³ with Jacobian determinant 1 which is not injective.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, MvPolynomial.RegularFunction, Fin, OfNat.ofNat, Nat
Proof uses: Matrix.vecCons, MvPolynomial, HAdd.hAdd, Finsupp, HMul.hMul, HPow.hPow, MvPolynomial.X, Neg.neg, Matrix.vecEmpty
41 instances, coercions and auxiliary constants left out.
```
noncomputable abbrev G [CommRing k] : RegularFunction k (Fin 3) (Fin 3) :=
…
```
lemma JacobianConjecture.det_jacobian_F line 55 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, Eq, MvPolynomial, Fin, OfNat.ofNat, Nat, Matrix.det, Finsupp, MvPolynomial.RegularFunction.Jacobian, JacobianConjecture.F, Neg.neg
Proof uses: Eq.mpr, HSub.hSub, HAdd.hAdd, HMul.hMul, MvPolynomial.X, RingHom, MvPolynomial.C, HPow.hPow, id, congr, congrArg, Eq.trans, Equiv, Matrix, Matrix.of, Derivation, MvPolynomial.pderiv, Matrix.vecCons, Matrix.vecEmpty, Matrix.det.congr_simp and 104 more
143 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma JacobianConjecture.det_jacobian_G line 68 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, Eq, MvPolynomial, Fin, OfNat.ofNat, Nat, Matrix.det, Finsupp, MvPolynomial.RegularFunction.Jacobian, JacobianConjecture.G
Proof uses: Eq.mpr, HSub.hSub, HAdd.hAdd, HMul.hMul, HPow.hPow, RingHom, MvPolynomial.C, MvPolynomial.X, Neg.neg, id, congrFun', congrArg, Eq.trans, Equiv, Matrix, Matrix.of, Derivation, MvPolynomial.pderiv, Matrix.vecCons, Matrix.vecEmpty and 94 more
97 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma JacobianConjecture.aeval_F_eq F identifies the two distinct points (0, 0, -1/4) and (1, -3/2, 13/2) line 78 source
Source documentation
F identifies the two distinct points (0, 0, -1/4) and (1, -3/2, 13/2).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, CharZero, Eq, Fin, OfNat.ofNat, Nat, MvPolynomial.RegularFunction.aeval, JacobianConjecture.F, Matrix.vecCons, HDiv.hDiv, Neg.neg, Matrix.vecEmpty
Proof uses: funext, List, List.Mem, List.finRange, HEq, Membership.mem, Finset, Fintype.elems, Fintype.complete, List.cons, List.Mem.head, Fin.mk, Nat.le_of_lt, Nat.le_refl, List.nil, Eq.mpr, HAdd.hAdd, HMul.hMul, HPow.hPow, id and 37 more
133 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
```
lemma aeval_F_eq [Field k] [CharZero k] :
    (F k).aeval ![(0 : k), 0, -1/4] = (F k).aeval ![(1 : k), -3/2, 13/2]  := by
…
```
Mentions: JacobianConjecture.F
lemma JacobianConjecture.aeval_G_eq G identifies the two distinct points (1, 0, 1) and (0, 3, -71) line 85 source
Source documentation
G identifies the two distinct points (1, 0, 1) and (0, 3, -71).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, Eq, Fin, OfNat.ofNat, Nat, MvPolynomial.RegularFunction.aeval, JacobianConjecture.G, Matrix.vecCons, Matrix.vecEmpty, Neg.neg
Proof uses: funext, List, List.Mem, List.finRange, HEq, Membership.mem, Finset, Fintype.elems, Fintype.complete, List.cons, List.Mem.head, Fin.mk, Nat.le_of_lt, Nat.le_refl, List.nil, Eq.mpr, HAdd.hAdd, HMul.hMul, HPow.hPow, id and 40 more
123 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
```
lemma aeval_G_eq [CommRing k] :
    (G k).aeval ![1, 0, (1 : k)] = (G k).aeval ![0, 3, -71] := by
…
```
Mentions: JacobianConjecture.G
def JacobianConjecture.JacobianConjectureProp The predicate that the Jacobian conjecture holds for a given field and variable index type (i.e. number of variables) line 92 source
Source documentation
The predicate that the Jacobian conjecture holds for a given field and variable index type (i.e. number of variables).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, Fintype, DecidableEq
Proof uses: MvPolynomial.RegularFunction, IsUnit, MvPolynomial, Finsupp, Nat, Matrix.det, MvPolynomial.RegularFunction.Jacobian, Exists, And, Eq, MvPolynomial.RegularFunction.comp, MvPolynomial.RegularFunction.id
11 instances, coercions and auxiliary constants left out.
```
def JacobianConjectureProp (k σ : Type*) [CommRing k] [Fintype σ] [DecidableEq σ] : Prop :=
…
```
theorem JacobianConjecture.jacobian_conjecture Documented declaration: jacobian_conjecture line 106 source
Source documentation
The Jacobian Conjecture: any regular function (i.e. vector valued polynomial function from) kⁿ → kᵐ whose Jacobian is a non-zero constant has an inverse that is given by a regular function, where k is a field of characteristic 0.
This is false: F has Jacobian determinant 1 but identifies two distinct points, so it admits no inverse. This counterexample works in all characteristics.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, Nontrivial, Iff, False, Fintype, DecidableEq, JacobianConjecture.JacobianConjectureProp
Proof uses: Eq.mpr, Not, id, Eq, congrArg, false_iff, MvPolynomial.RegularFunction, Fin, OfNat.ofNat, Nat, And, MvPolynomial.RegularFunction.comp, JacobianConjecture.G, MvPolynomial.RegularFunction.id, Exists, Eq.rec, MvPolynomial, Finsupp, IsUnit, isUnit_one and 24 more
40 instances, coercions and auxiliary constants left out.

Tactic trace: 9 steps
Open to load the checked before-and-after goal states.
```
theorem jacobian_conjecture {k : Type} [CommRing k] [Nontrivial k] :
    answer(False) ↔ ∀ {σ : Type} [Fintype σ] [DecidableEq σ], JacobianConjectureProp k σ := by
…
```
Mentions: JacobianConjecture.JacobianConjectureProp
theorem JacobianConjecture.jacobian_conjecture_two_variables Does the Jacobian conjecture hold in the two variable case? sorry line 120 source
Source documentation
Does the Jacobian conjecture hold in the two variable case?

depends on sorry what this means
Statement uses: Iff, True, Field, CharZero, JacobianConjecture.JacobianConjectureProp, Fin, OfNat.ofNat, Nat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
16 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem jacobian_conjecture_two_variables :
    answer(sorry) ↔ ∀ {k : Type} [Field k] [CharZero k], JacobianConjectureProp k (Fin 2) := by
…
```
Mentions: JacobianConjecture.JacobianConjectureProp
lemma JacobianConjecture.sanity_check_condition_1 Let's check that we've stated the "invertible Jacobian" condition correctly by proving an equivalence line 135 source
Source comment
Let's check that we've stated the "invertible Jacobian" condition correctly by proving an equivalence

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, DecidableEq, Field, MvPolynomial.RegularFunction, Iff, IsUnit, MvPolynomial, Finsupp, Nat, Matrix.det, MvPolynomial.RegularFunction.Jacobian, Exists, And, Ne, OfNat.ofNat, Eq, RingHom, MvPolynomial.C
Proof uses: of_eq_true, Not, Eq.trans, True, congrFun', congrArg, funext, iff_self
58 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
lemma sanity_check_condition_1 (F : RegularFunction k σ σ) :
    IsUnit F.Jacobian.det ↔ (∃ (c : k), c ≠ 0 ∧ F.Jacobian.det = .C c) := by
…
```
Mentions: JacobianConjecture.F, c
theorem JacobianConjecture.jacobian_conjecture_identity Let's apply the conjecture to a trivial case to make sure things are working as expected line 142 source
Source comment
Let's apply the conjecture to a trivial case to make sure things are working as expected.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, DecidableEq, Field, JacobianConjecture.JacobianConjectureProp, Exists, MvPolynomial.RegularFunction, And, Eq, MvPolynomial.RegularFunction.comp, MvPolynomial.RegularFunction.id
Proof uses: Matrix, MvPolynomial, MvPolynomial.RegularFunction.Jacobian, OfNat.ofNat, Finsupp, Nat, Matrix.ext, MvPolynomial.ext, of_eq_true, MvPolynomial.coeff, Eq.trans, Pi.single, True, congr, congrArg, congrFun', Equiv, Matrix.of, Derivation, MvPolynomial.pderiv and 9 more
34 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
```
theorem jacobian_conjecture_identity (H : JacobianConjectureProp k σ) :
    ∃ (G : RegularFunction k σ σ), G.comp (id k σ) = id k σ ∧
    (id k σ).comp G = id k σ := by
…
```
Mentions: JacobianConjecture.JacobianConjectureProp, JacobianConjecture.G