Tischler (1989): optimal constant for polynomials with only real roots

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SmaleMeanValue.tischler_real_roots

by Lucas · Sep 30, 2026 · Mathlib 0df444a (Lean v4.33.1)

complex-analysispolynomials

Let $P$ be a complex polynomial of degree $d \ge 2$ all of whose roots are real, and let $z \in \mathbb C$ with $P'(z) \neq 0$ (the point $z$ need not be real). Then there is a critical point $c$ of $P$ with

\left| \frac{P(z) - P(c)}{z - c} \right| \le \frac{d-1}{d}\,|P'(z)|.

This is the optimal form of Smale's conjecture for this class of polynomials; by the example $z^d - dz$ the constant cannot be lowered in general.

Preamble

import Mathlib
open Polynomial

Formal statement

namespace SmaleMeanValue

theorem tischler_real_roots (P : ℂ[X]) (hP : 2 ≤ P.natDegree)
    (hreal : ∀ r ∈ P.roots, r.im = 0) (z : ℂ) (hz : P.derivative.eval z ≠ 0) :
    ∃ c : ℂ, P.derivative.eval c = 0 ∧
      ‖(P.eval z - P.eval c) / (z - c)‖ ≤
        (((P.natDegree : ℝ) - 1) / P.natDegree) * ‖P.derivative.eval z‖ := by sorry

end SmaleMeanValue

Source

Wikipedia, "Mean value problem", revision oldid=1374678764, https://en.wikipedia.org/w/index.php?title=Mean_value_problem&oldid=1374678764, section "Partial results" (Tischler 1989, only real roots); Tischler, J. Complexity 5(4) (1989), 438–456, https://doi.org/10.1016/0885-064X(89)90019-8

Read-back

What the Lean code literally says, in plain math · Aristotle (Harmonic) — same agent as the drafter; NON-BLIND

Disclosure — NOT an independent read-back. This read-back is non-blind: it was written by the same agent that drafted the Lean statement, with full knowledge of the source material and of the intended meaning. It must not be treated as independent testimony. Reviewers should compare the Lean code against the source themselves.

Let $P$ be any polynomial with complex coefficients, and write $d$ for its degree (the natural-number degree; for the zero polynomial this would be $0$ , but here it is excluded because $d \ge 2$ is assumed). Write $P'$ for the formal derivative of $P$ and $P(w)$ for evaluation at $w \in \mathbb C$ . Assume every root of $P$ in $\mathbb C$ (taken from the multiset of roots, i.e. with multiplicity) has imaginary part $0$ , i.e. all complex roots of $P$ are real numbers. (Over $\mathbb C$ a polynomial of degree $d \ge 2$ has $d$ roots with multiplicity, so this hypothesis is not vacuous; it does not require the coefficients of $P$ to be real, e.g. $P = i(w-1)(w-2)$ qualifies.) Let $z$ be any complex number such that $P'(z) \neq 0$ . The point $z$ is not required to be real. The statement asserts that there exists $c \in \mathbb C$ with $P'(c) = 0$ and

\left| \frac{P(z) - P(c)}{z - c} \right| \le \frac{d-1}{d}\,|P'(z)| ,

where $\frac{d-1}{d}$ is computed in the real numbers from $d$ . Here $c$ ranges over all complex numbers; the condition $P'(c) = 0$ says that $c$ is a critical point of $P$ . Because $P'(z) \neq 0$ and $P'(c) = 0$ , necessarily $c \neq z$ , so the denominator $z - c$ is never zero and no division-by-zero convention is involved. The absolute values are the usual modulus on $\mathbb C$ .

Human review

Endorsed by Shuze Chen · Sep 30, 2026
Confirmed by the moderator at approval.
Endorsed by Lucas · Sep 30, 2026
Confirmed by the mission captain (proposal self-audit).

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