1043.lean`

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

3 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 1043

References:

erdosproblems.com/1043
[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.
[Po61] Pommerenke, Ch., On metric properties of complex polynomials. Michigan Math. J. (1961), 97-115.

Imports (1)

FormalConjecturesUtil

Imported by 1 file

FormalConjectures/ErdosProblems/1048.lean

def Erdos1043.levelSet The set $\{ z \in \mathbb{C} : \lvert f(z)\rvert\leq 1\}$ line 40 source
Source documentation
The set $\{ z \in \mathbb{C} : \lvert f(z)\rvert\leq 1\}$

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 levelSet (f : Polynomial ℂ) : Set ℂ :=
…
```
theorem Erdos1043.erdos_1043 sorry line 60 source
depends on sorry what this means
Statement uses: Iff, False, Polynomial, Complex, Polynomial.Monic, GE.ge, WithBot, Nat, Polynomial.degree, OfNat.ofNat, Exists, And, Eq, Real, Norm.norm, LE.le, ENNReal, MeasureTheory.Measure, Subtype, Membership.mem and 10 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
142 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos1043.erdos_1043.variants.weak On the other hand, Pommerenke also proved there always exists a line such that the projection has measure at most 3.3 sorry line 71 source
Source documentation
On the other hand, Pommerenke also proved there always exists a line such that the projection has measure at most 3.3.

depends on sorry what this means
Statement uses: Polynomial, Complex, Polynomial.Monic, GE.ge, WithBot, Nat, Polynomial.degree, OfNat.ofNat, Exists, And, Eq, Real, Norm.norm, LE.le, ENNReal, MeasureTheory.Measure, Subtype, Membership.mem, Submodule, Submodule.span and 10 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
133 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1043.variants.weak :
    ∀ (f : ℂ[X]), f.Monic → f.degree ≥ 1 →
      ∃ (u : ℂ), ‖u‖ = 1 ∧
      volume ((ℝ ∙ u).orthogonalProjectionOnto '' levelSet f) ≤ 3.3 := by
…
```
Mentions: Erdos1043.levelSet