1047.lean`

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

9 declarations, 6 contain sorry, 0 declared as axioms

Erdős Problem 1047

References:

erdosproblems.com/1047
[EHP58] Erdős, P. and Herzog, F. and Piranian, G., Metric properties of polynomials. J. Analyse Math. (1958), 125-148.
[Go66] Goodman, A. W., On the convexity of the level curves of a polynomial. Proc. Amer. Math. Soc. (1966), 358-361.
[Po61] Pommerenke, Ch., On metric properties of complex polynomials. Michigan Math. J. (1961), 97-115.

Imports (1)

FormalConjecturesUtil

def Erdos1047.sublevelSet The sublevel set $\{ z: \lvert f(z)\rvert\leq c\}$ of a polynomial $f\in\mathbb{C}[x]$ line 40 source
Source documentation
The sublevel set $\{ z: \lvert f(z)\rvert\leq c\}$ of a polynomial $f\in\mathbb{C}[x]$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Polynomial, Complex, Real, Set
Proof uses: Set.ofPred, LE.le, Norm.norm, Polynomial.eval
4 instances, coercions and auxiliary constants left out.
```
def sublevelSet (f : ℂ[X]) (c : ℝ) : Set ℂ := {z : ℂ | ‖f.eval z‖ ≤ c}
```
Mentions: c
def Erdos1047.strictSublevelSet The strict sublevel set $\{ z: \lvert f(z)\rvert< c\}$ of a polynomial $f\in\mathbb{C}[x]$ line 43 source
Source documentation
The strict sublevel set $\{ z: \lvert f(z)\rvert< c\}$ of a polynomial $f\in\mathbb{C}[x]$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Polynomial, Complex, Real, Set
Proof uses: Set.ofPred, LT.lt, Norm.norm, Polynomial.eval
4 instances, coercions and auxiliary constants left out.
```
def strictSublevelSet (f : ℂ[X]) (c : ℝ) : Set ℂ := {z : ℂ | ‖f.eval z‖ < c}
```
Mentions: c
def Erdos1047.componentsIn The connected components of a set s ⊆ ℂ, viewed as subsets of ℂ line 46 source
Source documentation
The connected components of a set s ⊆ ℂ, viewed as subsets of ℂ.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, Complex
Proof uses: Set.ofPred, Exists, And, Membership.mem, Eq, connectedComponentIn
8 instances, coercions and auxiliary constants left out.
```
def componentsIn (s : Set ℂ) : Set (Set ℂ) :=
…
```
theorem Erdos1047.erdos_1047 sorry line 67 source
depends on sorry what this means
Statement uses: Iff, False, Polynomial, Complex, Nat, Real, Polynomial.Monic, Eq, Set.ncard, Polynomial.rootSet, LT.lt, OfNat.ofNat, Set, Erdos1047.componentsIn, Erdos1047.sublevelSet, Membership.mem, Convex
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
59 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos1047.erdos_1047.variants.pommerenke Documented declaration: pommerenke sorry line 80 source
Source documentation
The answer is no, as shown by Pommerenke [Po61], who showed that, if $k$ is sufficiently large, and $f(z)=z^k(z-a)$ where $a$ is sufficiently close to $(1+\frac{1}{k})k^{\frac{1}{k+1}}$, then $\{ z: \lvert f(z)\rvert\leq 1\}$ has two components, and the component which contains $0$ is not convex.

depends on sorry what this means
Statement uses: Filter.Eventually, Nat, Exists, Real, And, LT.lt, OfNat.ofNat, HMul.hMul, HAdd.hAdd, HDiv.hDiv, HPow.hPow, Eq, Set.ncard, Set, Complex, Erdos1047.componentsIn, Erdos1047.sublevelSet, Polynomial, Polynomial.X, HSub.hSub and 6 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
104 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1047.variants.pommerenke :
    ∀ᶠ k : ℕ in Filter.atTop, ∃ δ : ℝ, 0 < δ ∧ ∀ a : ℝ,
      (1 + 1 / (k : ℝ)) * (k : ℝ) ^ ((1 : ℝ) / ((k : ℝ) + 1)) < a →
        a < (1 + 1 / (k : ℝ)) * (k : ℝ) ^ ((1 : ℝ) / ((k : ℝ) + 1)) + δ →
          (componentsIn (sublevelSet (X ^ k * (X - C (a : ℂ))) 1)).ncard = 2 ∧
          ¬ Convex ℝ (connectedComponentIn (sublevelSet (X ^ k * (X - C (a : ℂ))) 1) 0) := by
…
```
Mentions: Erdos1047.componentsIn, Erdos1047.sublevelSet
theorem Erdos1047.erdos_1047.variants.goodman Goodman [Go66] proved that one of the three components of $$\{ z: \lvert (z^2+1)(z-2)^2\rvert < 5^{3/2}/4\}$$ is not convex sorry line 94 source
Source documentation
Goodman [Go66] proved that one of the three components of $$\{ z: \lvert (z^2+1)(z-2)^2\rvert < 5^{3/2}/4\}$$ is not convex.

depends on sorry what this means
Statement uses: And, Eq, Nat, Set.ncard, Set, Complex, Erdos1047.componentsIn, Erdos1047.strictSublevelSet, HMul.hMul, Polynomial, HAdd.hAdd, HPow.hPow, Polynomial.X, OfNat.ofNat, HSub.hSub, RingHom, Polynomial.C, HDiv.hDiv, Real, Exists and 3 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
92 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1047.variants.goodman :
    (componentsIn (strictSublevelSet ((X ^ 2 + 1) * (X - C 2) ^ 2)
        ((5 : ℝ) ^ ((3 : ℝ) / 2) / 4))).ncard = 3 ∧
      ∃ t ∈ componentsIn (strictSublevelSet ((X ^ 2 + 1) * (X - C 2) ^ 2)
        ((5 : ℝ) ^ ((3 : ℝ) / 2) / 4)), ¬ Convex ℝ t := by
…
```
Mentions: Erdos1047.componentsIn, Erdos1047.strictSublevelSet
theorem Erdos1047.erdos_1047.variants.goodman_simple_roots Goodman [Go66] constructed an example with simple roots, of degree $4$ sorry line 105 source
Source documentation
Goodman [Go66] constructed an example with simple roots, of degree $4$.

depends on sorry what this means
Statement uses: Exists, Polynomial, Complex, Real, And, Polynomial.Monic, Eq, Nat, Polynomial.natDegree, OfNat.ofNat, Set.ncard, Polynomial.rootSet, LT.lt, Set, Erdos1047.componentsIn, Erdos1047.sublevelSet, Membership.mem, Not, Convex
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
60 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1047.variants.goodman_simple_roots :
    ∃ (f : ℂ[X]) (c : ℝ), f.Monic ∧ f.natDegree = 4 ∧ (f.rootSet ℂ).ncard = 4 ∧ 0 < c ∧
      (componentsIn (sublevelSet f c)).ncard = 4 ∧
      ∃ t ∈ componentsIn (sublevelSet f c), ¬ Convex ℝ t := by
…
```
Mentions: c, Erdos1047.componentsIn, Erdos1047.sublevelSet
theorem Erdos1047.erdos_1047.variants.referee The referee of the paper [Go66] also gave the example of $\lvert z(z^5-1)\rvert< 5\cdot 6^{-6/5}$ sorry line 115 source
Source documentation
The referee of the paper [Go66] also gave the example of $\lvert z(z^5-1)\rvert< 5\cdot 6^{-6/5}$.

depends on sorry what this means
Statement uses: And, Eq, Nat, Set.ncard, Set, Complex, Erdos1047.componentsIn, Erdos1047.strictSublevelSet, HMul.hMul, Polynomial, Polynomial.X, HSub.hSub, HPow.hPow, OfNat.ofNat, Real, HDiv.hDiv, Neg.neg, Exists, Membership.mem, Not and 1 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
84 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1047.variants.referee :
    (componentsIn (strictSublevelSet (X * (X ^ 5 - 1))
        (5 * (6 : ℝ) ^ (-(6 : ℝ) / 5)))).ncard = 6 ∧
      ∃ t ∈ componentsIn (strictSublevelSet (X * (X ^ 5 - 1))
        (5 * (6 : ℝ) ^ (-(6 : ℝ) / 5))), ¬ Convex ℝ t := by
…
```
Mentions: Erdos1047.componentsIn, Erdos1047.strictSublevelSet
theorem Erdos1047.erdos_1047.variants.max_non_convex_components sorry line 131 source
depends on sorry what this means
Statement uses: Nat, IsGreatest, Set.ofPred, Exists, Polynomial, Complex, Real, And, Polynomial.Monic, Eq, Polynomial.natDegree, LT.lt, OfNat.ofNat, Set.ncard, Set, Membership.mem, Erdos1047.componentsIn, Erdos1047.strictSublevelSet, Not, Convex and 2 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
52 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.