CasasAlvero.lean`

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

10 declarations, 4 contain sorry, 0 declared as axioms

Casas-Alvero Conjecture

References:

The Casas-Alvero conjecture states that if a univariate polynomial P of degree d over a field of characteristic zero shares a non-trivial factor with its Hasse derivatives up to order d-1, then P must be of the form (X - α)ᵈ for some α in the field.

The conjecture has been proven for:

Degrees d ≤ 8
Degrees of the form p^k where p is prime
Degrees of the form 2p^k where p is prime

The conjecture is false in positive characteristic p for polynomials of degree p+1.

The conjecture is now claimed to be proven in this paper:

Proof of the Casas-Alvero conjecture: Soham Ghosh)

Imports (1)

FormalConjecturesUtil

def CasasAlvero.HasCasasAlveroProp A polynomial P satisfies the Casas-Alvero property if it shares a factor with each of its Hasse derivatives up to order d-1, where d is the degree of P line 55 source
Source documentation
A polynomial P satisfies the Casas-Alvero property if it shares a factor with each of its Hasse derivatives up to order d-1, where d is the degree of P.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, Polynomial
Proof uses: Nat, Membership.mem, Finset, Finset.range, Polynomial.natDegree, Not, IsCoprime, LinearMap, RingHom.id, Polynomial.hasseDeriv
17 instances, coercions and auxiliary constants left out.
```
def HasCasasAlveroProp (P : K[X]) : Prop :=
…
```
def CasasAlvero.HasCasasAlveroPropᵣ A stronger version of the Casas-Alvero property, which requires that the polynomial P shares a root with each of its Hasse derivatives up to order deg P - 1 line 62 source
Source documentation
A stronger version of the Casas-Alvero property, which requires that the polynomial P shares a root with each of its Hasse derivatives up to order deg P - 1. The subscript r indicates "root" in the definition.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, Polynomial
Proof uses: Nat, Membership.mem, Finset, Finset.range, Polynomial.natDegree, Exists, And, Polynomial.IsRoot, LinearMap, RingHom.id, Polynomial.hasseDeriv
15 instances, coercions and auxiliary constants left out.
```
def HasCasasAlveroPropᵣ (P : K[X]) : Prop :=
…
```
theorem CasasAlvero.HasCasasAlveroPropᵣ.hasCasasAlveroProp line 66 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, Polynomial, CasasAlvero.HasCasasAlveroPropᵣ, CasasAlvero.HasCasasAlveroProp
Proof uses: Nat, Membership.mem, Finset, Finset.range, Polynomial.natDegree, IsCoprime, LinearMap, RingHom.id, Polynomial.hasseDeriv, Exists, And, Dvd.dvd, HSub.hSub, Polynomial.X, RingHom, Polynomial.C, False, Eq.mp, Polynomial.IsRoot, forall_congr and 7 more
44 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
theorem CasasAlvero.HasCasasAlveroProp.map_iff line 74 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, RingHom, Polynomial, Iff, CasasAlvero.HasCasasAlveroProp, Polynomial.map
Proof uses: of_eq_true, Eq.trans, Nat, LT.lt, Polynomial.natDegree, Not, IsCoprime, LinearMap, RingHom.id, Polynomial.hasseDeriv, True, congr, congrArg, forall_congr, Membership.mem, Finset, Finset.range, implies_congr, congrFun', Polynomial.natDegree_map and 3 more
35 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem CasasAlvero.hasCasasAlveroProp_iffᵣ line 79 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, Polynomial, IsAlgClosed, Iff, CasasAlvero.HasCasasAlveroProp, CasasAlvero.HasCasasAlveroPropᵣ
Proof uses: Iff.intro, id, Nat, Membership.mem, Finset, Finset.range, Polynomial.natDegree, Not, IsCoprime, LinearMap, RingHom.id, Polynomial.hasseDeriv, Eq.mpr, Exists, And, Polynomial.IsRoot, Or, Ne, AlgHom, Polynomial.aeval and 14 more
26 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
lemma CasasAlvero.casas_alvero_iffᵣ Note that whether we use HasCasasAlveroProp or HasCasasAlveroPropᵣ to state the Casas-Alvero conjecture, we obtain the following equivalent statements line 93 source
Source documentation
Note that whether we use HasCasasAlveroProp or HasCasasAlveroPropᵣ to state the Casas-Alvero conjecture, we obtain the following equivalent statements.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Iff, Field, CharZero, Polynomial, Polynomial.Monic, CasasAlvero.HasCasasAlveroProp, Exists, Eq, HPow.hPow, Nat, HSub.hSub, Polynomial.X, RingHom, Polynomial.C, Polynomial.natDegree, CasasAlvero.HasCasasAlveroPropᵣ
Proof uses: Iff.intro, CasasAlvero.HasCasasAlveroPropᵣ.hasCasasAlveroProp, AlgebraicClosure, Polynomial.map, Algebra.algebraMap, Polynomial.Monic.map, Iff.mp, CasasAlvero.hasCasasAlveroProp_iffᵣ, Iff.mpr, CasasAlvero.HasCasasAlveroProp.map_iff, dite, OfNat.ofNat, of_eq_true, Eq.trans, True, congrArg, funext, congr, Polynomial.Monic.natDegree_eq_zero, Polynomial.natDegree_one and 42 more
140 instances, coercions and auxiliary constants left out.

Tactic trace: 13 steps
Open to load the checked before-and-after goal states.
```
lemma casas_alvero_iffᵣ :
    (∀ {K : Type u} [Field K] [CharZero K] (P : K[X]),
      P.Monic → HasCasasAlveroProp P → ∃ α : K, P = (X - C α) ^ P.natDegree) ↔
    (∀ {K : Type u} [Field K] [CharZero K] (P : K[X]),
      P.Monic → HasCasasAlveroPropᵣ P → ∃ α : K, P = (X - C α) ^ P.natDegree) := by
…
```
Mentions: CasasAlvero.HasCasasAlveroProp, CasasAlvero.HasCasasAlveroPropᵣ
theorem CasasAlvero.casas_alvero_conjecture The Casas-Alvero conjecture states that in characteristic zero, if a monic polynomial P has the Casas-Alvero property, then P = (X - α)ᵈ for some α sorry line 124 source
Source documentation
The Casas-Alvero conjecture states that in characteristic zero, if a monic polynomial P has the Casas-Alvero property, then P = (X - α)ᵈ for some α.

depends on sorry what this means
Statement uses: Field, CharZero, Polynomial, Polynomial.Monic, CasasAlvero.HasCasasAlveroProp, Exists, Eq, HPow.hPow, Nat, HSub.hSub, Polynomial.X, RingHom, Polynomial.C, Polynomial.natDegree
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
35 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem casas_alvero_conjecture (hP' : HasCasasAlveroProp P) :
    ∃ α : K, P = (X - C α) ^ P.natDegree := by
…
```
Mentions: CasasAlvero.HasCasasAlveroProp
theorem CasasAlvero.casas_alvero.prime_power The Casas-Alvero conjecture holds for polynomials of prime power degree sorry line 135 source
Source documentation
The Casas-Alvero conjecture holds for polynomials of prime power degree. This was proved by Graf von Bothmer, Labs, Schicho, and van de Woestijne.
Reference: The Casas-Alvero conjecture for infinitely many degrees

L4DEL explanation
The Casas-Alvero conjecture holds for polynomials of prime power degree. This was proved by Graf von Bothmer, Labs, Schicho, and van de Woestijne. Reference: The Casas-Alvero conjecture for infinitely many degrees
Sources and attribution: Pinned Lean source (1646ca16afd6)

depends on sorry what this means
Statement uses: Field, CharZero, Polynomial, Polynomial.Monic, Nat, Nat.Prime, Eq, Polynomial.natDegree, HPow.hPow, CasasAlvero.HasCasasAlveroProp, Exists, HSub.hSub, Polynomial.X, RingHom, Polynomial.C
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
37 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem casas_alvero.prime_power (p k : ℕ) (hp : p.Prime) (hd : P.natDegree = p^k)
    (hP' : HasCasasAlveroProp P) : ∃ α : K, P = (X - C α) ^ P.natDegree := by
…
```
Mentions: CasasAlvero.HasCasasAlveroProp
theorem CasasAlvero.casas_alvero.double_prime_power The Casas-Alvero conjecture holds for polynomials of degree 2p^k where p is prime sorry line 146 source
Source documentation
The Casas-Alvero conjecture holds for polynomials of degree 2p^k where p is prime. This was proved by Graf von Bothmer, Labs, Schicho, and van de Woestijne.
Reference: The Casas-Alvero conjecture for infinitely many degrees

L4DEL explanation
The Casas-Alvero conjecture holds for polynomials of degree 2p^k where p is prime. This was proved by Graf von Bothmer, Labs, Schicho, and van de Woestijne. Reference: The Casas-Alvero conjecture for infinitely many degrees
Sources and attribution: Pinned Lean source (1646ca16afd6)

depends on sorry what this means
Statement uses: Field, CharZero, Polynomial, Polynomial.Monic, Nat, Nat.Prime, Eq, Polynomial.natDegree, HMul.hMul, OfNat.ofNat, HPow.hPow, CasasAlvero.HasCasasAlveroProp, Exists, HSub.hSub, Polynomial.X, RingHom, Polynomial.C
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
42 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem casas_alvero.double_prime_power (p k : ℕ) (hp : p.Prime) (hd : P.natDegree = 2 * p^k)
    (hP' : HasCasasAlveroProp P) : ∃ α : K, P = (X - C α) ^ P.natDegree := by
…
```
Mentions: CasasAlvero.HasCasasAlveroProp
theorem CasasAlvero.casas_alvero.positive_char_counterexample The Casas-Alvero conjecture fails in positive characteristic p for polynomials of degree p + 1 sorry line 161 source
Source documentation
The Casas-Alvero conjecture fails in positive characteristic p for polynomials of degree p + 1. This was shown by Graf von Bothmer, Labs, Schicho, and van de Woestijne.
Reference: The Casas-Alvero conjecture for infinitely many degrees
Formal proof linked here provided by AlphaProof.

L4DEL explanation
The Casas-Alvero conjecture fails in positive characteristic p for polynomials of degree p + 1. This was shown by Graf von Bothmer, Labs, Schicho, and van de Woestijne. Reference: The Casas-Alvero conjecture for infinitely many degrees Formal proof linked here provided by AlphaProof.
Sources and attribution: Pinned Lean source (1646ca16afd6)

depends on sorry what this means
Statement uses: Nat, Nat.Prime, Exists, Field, CharP, Polynomial, HSub.hSub, HPow.hPow, Polynomial.X, HAdd.hAdd, OfNat.ofNat, And, Polynomial.Monic, CasasAlvero.HasCasasAlveroProp, Not, Eq, RingHom, Polynomial.C, Polynomial.natDegree
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
44 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem casas_alvero.positive_char_counterexample {p : ℕ} (hp : p.Prime) :
    ∃ (K : Type*) (_ : Field K) (_ : CharP K p),
      let P := X ^ (p + 1) - X ^ p
…
```