1046.lean`

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

5 declarations, 3 contain sorry, 0 declared as axioms

Erdős Problem 1046

References:

erdosproblems.com/1046
[EHP58] Erdős, P. and Herzog, F. and Piranian, G., Metric properties of polynomials. J. Analyse Math. (1958), 125-148.
[Po59] Pommerenke, Ch., On some problems by Erdős, Herzog and Piranian. Michigan Math. J. (1959), 221-225.
[BPS02] Barnard, Roger W. and Pearce, Kent and Solynin, Alexander Yu., An isoperimetric inequality for logarithmic capacity. Ann. Acad. Sci. Fenn. Math. (2002), 419-436.
[Po28] Pólya, G., Beitrag zur Verallgemeinerung des Verzerrungssatzes auf mehrfach zusammenhängende Gebiete. S.-B. Preuss. Akad. Wiss. (1928), 280-282.
[Ra95] Ransford, Thomas, Potential theory in the complex plane. London Mathematical Society Student Texts 28, Cambridge University Press (1995).

Imports (1)

FormalConjecturesUtil

def Erdos1046.lemniscate The open unit lemniscate $E = \{z : \lvert f(z)\rvert < 1\}$ of a complex polynomial line 44 source
Source documentation
The open unit lemniscate $E = \{z : \lvert f(z)\rvert < 1\}$ of a complex polynomial.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Polynomial, Complex, Set
Proof uses: Set.ofPred, LT.lt, Real, Norm.norm, Polynomial.eval, OfNat.ofNat
6 instances, coercions and auxiliary constants left out.
```
def lemniscate (f : ℂ[X]) : Set ℂ := {z | ‖f.eval z‖ < 1}
```
def Erdos1046.closedLemniscate The closed unit lemniscate $\{z : \lvert f(z)\rvert \leq 1\}$ of a complex polynomial line 47 source
Source documentation
The closed unit lemniscate $\{z : \lvert f(z)\rvert \leq 1\}$ of a complex polynomial.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Polynomial, Complex, Set
Proof uses: Set.ofPred, LE.le, Real, Norm.norm, Polynomial.eval, OfNat.ofNat
6 instances, coercions and auxiliary constants left out.
```
def closedLemniscate (f : ℂ[X]) : Set ℂ := {z | ‖f.eval z‖ ≤ 1}
…
```
theorem Erdos1046.erdos_1046 sorry line 67 source
depends on sorry what this means
Statement uses: Iff, True, Polynomial, Complex, Polynomial.Monic, IsConnected, Erdos1046.lemniscate, Exists, LE.le, Set, Metric.ball, OfNat.ofNat, Real, Nat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
28 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos1046.erdos_1046.variants.centroid sorry line 77 source
depends on sorry what this means
Statement uses: Polynomial, Complex, Polynomial.Monic, IsConnected, Erdos1046.lemniscate, LE.le, Set, Metric.ball, HDiv.hDiv, Multiset.sum, Polynomial.roots, Polynomial.natDegree, OfNat.ofNat, Real, Nat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
50 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos1046.erdos_1046.variants.diameter Erdős, Herzog, and Piranian [EHP58] conjecture that if $\{ z: \lvert f(z)\rvert\leq 1\}$ is connected then its diameter is at least $2$ sorry line 91 source
Source documentation
Erdős, Herzog, and Piranian [EHP58] conjecture that if $\{ z: \lvert f(z)\rvert\leq 1\}$ is connected then its diameter is at least $2$.
The answer is yes, and connectedness is not needed. For monic $f$ of degree $n \geq 1$ the set $\{ z: \lvert f(z)\rvert\leq 1\}$ has logarithmic capacity $1$ [Ra95, Theorem 5.2.5], and Pólya [Po28] proved $\mathrm{cap}(E) \leq \frac{1}{2}\operatorname{diam}(E)$ for every compact $E \subseteq \mathbb{C}$, see [BPS02, (1.2)]. Hence the diameter is at least $2$.

depends on sorry what this means
Statement uses: Iff, True, Polynomial, Complex, Polynomial.Monic, IsConnected, Erdos1046.closedLemniscate, LE.le, ENNReal, OfNat.ofNat, Nat, Metric.ediam
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
40 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1046.variants.diameter : answer(True) ↔ ∀ f : ℂ[X], f.Monic →
    IsConnected (closedLemniscate f) → 2 ≤ Metric.ediam (closedLemniscate f) := by
…
```
Mentions: Erdos1046.closedLemniscate