114.lean`

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6 declarations, 3 contain sorry, 0 declared as axioms

Erdős Problem 114

References:

erdosproblems.com/114
[EHP58] Erdős, P., Herzog, F. and Piranian, G., Metric properties of polynomials. J. Analyse Math. (1958), 125-148.
[ErHa99] Eremenko, A. and Hayman, W., On the length of lemniscates. Michigan Math. J. (1999), 409-415. arXiv:0805.2295
[Ta25] Tao, T., The maximal length of the Erdős–Herzog–Piranian lemniscate in high degree (2025). arXiv:2512.12455

Imports (1)

FormalConjecturesUtil

def Erdos114.lemniscate The lemniscate $\{z \in \mathbb{C} : \lvert p(z)\rvert = 1\}$ of a complex polynomial $p$ line 40 source
Source documentation
The lemniscate $\{z \in \mathbb{C} : \lvert p(z)\rvert = 1\}$ of a complex polynomial $p$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Polynomial, Complex, Set
Proof uses: Set.ofPred, Eq, Real, Norm.norm, Polynomial.eval, OfNat.ofNat
5 instances, coercions and auxiliary constants left out.
```
def lemniscate (p : ℂ[X]) : Set ℂ := {z | ‖p.eval z‖ = 1}
```
def Erdos114.lemniscateLength The length of the lemniscate of $p$, as its one-dimensional Hausdorff measure line 43 source
Source documentation
The length of the lemniscate of $p$, as its one-dimensional Hausdorff measure.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Polynomial, Complex, ENNReal
Proof uses: MeasureTheory.Measure, Set, MeasureTheory.Measure.hausdorffMeasure, OfNat.ofNat, Real, Erdos114.lemniscate
11 instances, coercions and auxiliary constants left out.
```
noncomputable def lemniscateLength (p : ℂ[X]) : ENNReal := μH[1] (lemniscate p)
…
```
Mentions: Erdos114.lemniscate
theorem Erdos114.erdos_114 If $p(z) \in \mathbb{C}[z]$ is a monic polynomial of degree $n$, is the length of the curve $\{z \in \mathbb{C} : \lvert p(z)\rvert = 1\}$ maximised when $p(z) = z^n - 1$? sorry line 53 source
Source documentation
If $p(z) \in \mathbb{C}[z]$ is a monic polynomial of degree $n$, is the length of the curve $\{z \in \mathbb{C} : \lvert p(z)\rvert = 1\}$ maximised when $p(z) = z^n - 1$?
We require $n \geq 1$. For $n = 0$ the only monic polynomial is $1$, whose level set is all of $\mathbb{C}$, while $z^0 - 1 = 0$ has an empty level set.

depends on sorry what this means
Statement uses: Iff, True, Nat, LE.le, OfNat.ofNat, Polynomial, Complex, Polynomial.Monic, Eq, Polynomial.natDegree, ENNReal, Erdos114.lemniscateLength, HSub.hSub, HPow.hPow, Polynomial.X
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 erdos_114 : answer(sorry) ↔
    ∀ n : ℕ, 1 ≤ n → ∀ p : ℂ[X], p.Monic → p.natDegree = n →
      lemniscateLength p ≤ lemniscateLength (X ^ n - 1) := by
…
```
Mentions: Erdos114.lemniscateLength
theorem Erdos114.erdos_114.variants.large_n Tao [Ta25] proved that $z^n - 1$ maximises the length of the lemniscate for all sufficiently large $n$ sorry line 63 source
Source documentation
Tao [Ta25] proved that $z^n - 1$ maximises the length of the lemniscate for all sufficiently large $n$.

depends on sorry what this means
Statement uses: Filter.Eventually, Nat, Polynomial, Complex, Polynomial.Monic, Eq, Polynomial.natDegree, LE.le, ENNReal, Erdos114.lemniscateLength, HSub.hSub, HPow.hPow, Polynomial.X, OfNat.ofNat, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
27 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_114.variants.large_n : ∀ᶠ n in atTop, ∀ p : ℂ[X], p.Monic → p.natDegree = n →
    lemniscateLength p ≤ lemniscateLength (X ^ n - 1) := by
…
```
Mentions: Erdos114.lemniscateLength
theorem Erdos114.erdos_114.variants.two Eremenko and Hayman [ErHa99] proved the case $n = 2$ sorry line 69 source
Source documentation
Eremenko and Hayman [ErHa99] proved the case $n = 2$.

depends on sorry what this means
Statement uses: Polynomial, Complex, Polynomial.Monic, Eq, Nat, Polynomial.natDegree, OfNat.ofNat, LE.le, ENNReal, Erdos114.lemniscateLength, HSub.hSub, HPow.hPow, Polynomial.X
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
26 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_114.variants.two (p : ℂ[X]) (hp : p.Monic) (hdeg : p.natDegree = 2) :
    lemniscateLength p ≤ lemniscateLength (X ^ 2 - 1) := by
…
```
Mentions: Erdos114.lemniscateLength
theorem Erdos114.erdos_114.variants.one The case $n = 1$ line 78 source
Source documentation
The case $n = 1$. A monic linear polynomial is $z - a$. Its lemniscate is a translate of the lemniscate of $z - 1$, and translations preserve Hausdorff measure.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Polynomial, Complex, Polynomial.Monic, Eq, Nat, Polynomial.natDegree, OfNat.ofNat, LE.le, ENNReal, Erdos114.lemniscateLength, HSub.hSub, HPow.hPow, Polynomial.X
Proof uses: HAdd.hAdd, RingHom, Polynomial.C, Polynomial.coeff, Finset.sum, Finset.range, HMul.hMul, Polynomial.Monic.as_sum, Eq.mp, congrArg, congr, pow_one, Eq.trans, Singleton.singleton, Finset, Finset.sum_singleton, pow_zero, mul_one, Set, Erdos114.lemniscate and 68 more
96 instances, coercions and auxiliary constants left out.

Tactic trace: 10 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_114.variants.one (p : ℂ[X]) (hp : p.Monic) (hdeg : p.natDegree = 1) :
    lemniscateLength p ≤ lemniscateLength (X ^ 1 - 1) := by
…
```
Mentions: Erdos114.lemniscateLength