MeanValueProblem.lean`

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

4 declarations, 4 contain sorry, 0 declared as axioms

Mean value problem

Reference:

Wikipedia
The fundamental theorem of algebra and complexity theory by Steve Smale

Given a complex polynomial $p$ of degree $d ≥ 2$ and a complex number $z$ there is a critical point $c$ of $p$, such that $|p(z)-p(c)|/|z-c| ≤ K* |p'(z)|$ for $K=1$.

The conjecture has been proven for:

K = 4 The fundamental theorem of algebra and complexity theory by Steve Smale
K = (d-1)/d at a point $z$ if the normalised polynomial $p(X + z) - p(z)$ has only real roots, or if all its nonzero roots have the same norm. Critical points and values of complex polynomials by David Tischler

Imports (1)

FormalConjecturesUtil

lemma MeanValueProblem.mean_value_problem Given a complex polynomial $p$ of degree $d ≥ 2$ and a complex number $z$ there is a critical point $c$ of $p$, such that $|p(z)-p(c)|/|z-c| ≤ |p'(z)|$ sorry line 52 source
Source documentation
Given a complex polynomial $p$ of degree $d ≥ 2$ and a complex number $z$ there is a critical point $c$ of $p$, such that $|p(z)-p(c)|/|z-c| ≤ |p'(z)|$.

depends on sorry what this means
Statement uses: Polynomial, Complex, LE.le, WithBot, Nat, OfNat.ofNat, Polynomial.degree, Real, Exists, And, Eq, Polynomial.eval, LinearMap, RingHom.id, Polynomial.derivative, HDiv.hDiv, Norm.norm, HSub.hSub
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.
```
lemma mean_value_problem (p : Polynomial ℂ) (hp : 2 ≤ p.degree) (z : ℂ) (K : ℝ):
    ∃ c : ℂ, p.derivative.eval c = 0 ∧
      ‖p.eval z - p.eval c‖ / ‖z - c‖ ≤ ‖p.derivative.eval z‖ := by
…
```
Mentions: c
lemma MeanValueProblem.mean_value_problem_leq_4 The following weaker version of the mean value problem has been proven sorry line 63 source
Source documentation
The following weaker version of the mean value problem has been proven. Given a complex polynomial $p$ of degree $d ≥ 2$ and a complex number $z$ there a critical point $c$ of $p$, such that $|p(z)-p(c)|/|z-c| ≤ 4|p'(z)|$.

depends on sorry what this means
Statement uses: Polynomial, Complex, LE.le, WithBot, Nat, OfNat.ofNat, Polynomial.degree, Real, Exists, And, Eq, Polynomial.eval, LinearMap, RingHom.id, Polynomial.derivative, HDiv.hDiv, Norm.norm, HSub.hSub, HMul.hMul
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
58 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
lemma mean_value_problem_leq_4 (p : Polynomial ℂ) (hp : 2 ≤ p.degree) (z : ℂ) (K : ℝ):
    ∃ c : ℂ, p.derivative.eval c = 0 ∧
      ‖p.eval z - p.eval c‖ / ‖z - c‖ ≤ 4 *‖p.derivative.eval z‖ := by
…
```
Mentions: c
lemma MeanValueProblem.mean_value_problem_of_real_roots The following tighter bound depending on the degree $d$ of the polynomial $p$ has been shown by Tischler sorry line 77 source
Source documentation
The following tighter bound depending on the degree $d$ of the polynomial $p$ has been shown by Tischler. Let $z$ be a complex number such that the normalised polynomial $p(X + z) - p(z)$ has only real roots, i.e. every $x$ with $p(x) = p(z)$ satisfies $\operatorname{Im} x = \operatorname{Im} z$. Then there is a critical point $c$ of $p$ such that $|p(z)-p(c)|/|z-c| \le (d-1)/d \cdot |p'(z)|$.

depends on sorry what this means
Statement uses: Polynomial, Complex, LE.le, Nat, OfNat.ofNat, Polynomial.natDegree, Eq, Polynomial.eval, Real, Complex.im, Exists, And, LinearMap, RingHom.id, Polynomial.derivative, HDiv.hDiv, Norm.norm, HSub.hSub, HMul.hMul
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.
```
lemma mean_value_problem_of_real_roots (p : Polynomial ℂ) (hp : 2 ≤ p.natDegree) (z : ℂ)
    (h : ∀ x : ℂ, p.eval x = p.eval z → x.im = z.im) (K : ℝ) :
    ∃ c : ℂ, p.derivative.eval c = 0 ∧
      ‖p.eval z - p.eval c‖ / ‖z - c‖ ≤ (p.natDegree - 1)/ p.natDegree * ‖p.derivative.eval z‖ := by
…
```
Mentions: c
lemma MeanValueProblem.mean_value_problem_of_roots_same_norm The following tighter bound depending on the degree $d$ of the polynomial $p$ has been shown by Tischler sorry line 92 source
Source documentation
The following tighter bound depending on the degree $d$ of the polynomial $p$ has been shown by Tischler. Let $z$ be a complex number such that all nonzero roots of the normalised polynomial $p(X + z) - p(z)$ have the same norm, i.e. all $x \ne z$ with $p(x) = p(z)$ have the same distance to $z$. Then there is a critical point $c$ of $p$ such that $|p(z) - p(c)|/|z-c| \le (d-1)/d \cdot |p'(z)|$.

depends on sorry what this means
Statement uses: Polynomial, Complex, LE.le, Nat, OfNat.ofNat, Polynomial.natDegree, Eq, Polynomial.eval, Ne, Real, Norm.norm, HSub.hSub, Exists, And, LinearMap, RingHom.id, Polynomial.derivative, HDiv.hDiv, HMul.hMul
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.
```
lemma mean_value_problem_of_roots_same_norm (p : Polynomial ℂ) (hp : 2 ≤ p.natDegree) (z : ℂ)
    (h : ∀ x y : ℂ, p.eval x = p.eval z → p.eval y = p.eval z → x ≠ z → y ≠ z →
      ‖x - z‖ = ‖y - z‖) (K : ℝ) :
    ∃ c : ℂ, p.derivative.eval c = 0 ∧
      ‖p.eval z - p.eval c‖ / ‖z - c‖ ≤ (p.natDegree - 1)/ p.natDegree * ‖p.derivative.eval z‖ := by
…
```
Mentions: c