bibkey: krishna2022cstarsmale authors: K. Mahesh Krishna year: 2022 title: “C*-algebraic Smale Mean Value Conjecture and Dubinin-Sugawa Dual Mean Value Conjecture” doi: 10.48550/arXiv.2206.08154 url: https://arxiv.org/abs/2206.08154v1 claim: “Conjecture HIGHERMEAN (journal Conjecture 2.3) and Conjecture DUALSMALE (Conjecture 3.1) ask for a critical point controlling the Smale ratios, respectively the Dubinin-Sugawa dual ratio, in a commutative C*-algebra; the paper proves both degree-2 cases.” strata_touched:

• D5/S3/Quantum/Algebra/CStarSmaleHigherOrder
• D5/S3/Quantum/Algebra/CStarDualMeanValue license: citation-only triage: anchor

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DOI: 10.48550/arXiv.2206.08154

Source: https://arxiv.org/abs/2206.08154v1

Journal DOI: 10.26117/2079-6641-2026-54-2-38-47

Krishna, C*-algebraic Smale mean value conjectures

K. Mahesh Krishna, C-algebraic Smale Mean Value Conjecture and Dubinin-Sugawa Dual Mean Value Conjecture*, arXiv:2206.08154v1, Section 2. The journal version appears in Vestnik KRAUNC Fiz.-Mat. Nauki 54(2), 38–47, and labels HIGHERMEAN as Conjecture 2.3 on p. 44.

Source statement

Conjecture HIGHERMEAN (Higher Order C-algebraic Smale Mean Value Conjecture).*

Let $\mathcal{A}$ be a commutative C*-algebra. Let $P(z):=(z-a_1)\cdots (z-a_n)$ be a polynomial of degree $n\ge 2$ over $\mathcal{A}$, $a_1,\dots ,a_n\in \mathcal{A}$. If $z\in \mathcal{A}$ is not a critical point of $P$, then there exists a critical point $w\in \mathcal{A}$ of $P$ such that

$$>\frac{\Vert P^{(k)}(z)\Vert }{k!}\frac{\Vert P(z)-P(w){\Vert }^{k-1}}{\Vert P^{\prime }(z){\Vert }^k}>\le 4^{k-1},\forall 2\le k\le n.>$$

The following theorem states: “Conjecture 2.3 holds for degree 2 C*-algebraic polynomials.” In the journal this is Theorem 2.2, p. 44. The scalar Higher Order Smale Mean Value Theorem is Theorem 1.3 in the journal; the C*-algebraic extension is a distinct statement.

Encoding and scope

smalePoly is the literal product of the factors Polynomial.X - Polynomial.C (a j) for j : Fin n; the index j represents the source’s a_(j+1). The source’s sum-of-omitted-factors derivative is Polynomial.derivative_prod, and higher derivatives are iterates of Polynomial.derivative. The unital complex commutative C*-algebra convention is CommCStarAlgebra. A refutation on Complex × Complex suffices for the source’s universal claim, including any broader interpretation allowing non-unital algebras.

The degree-three refutation uses the supremum norm on Complex × Complex. It does not alter the degree-2 theorem or the scalar theorem. The paper’s CSMALE, its strong form and its dynamics conjecture are separate questions.

Conjecture DUALSMALE

Section 2, p. 4, defines the polynomial and its derivative:

Let 𝒜 be a C*-algebra. For P(z) ≔ (z−a₁)(z−a₂)⋯(z−aₙ) for all z∈𝒜 with a₁, a₂, …, aₙ ∈ 𝒜, we define P′(z)=∑ⱼ₌₁ⁿ (z−a₁)⋯(z−aⱼ)̂⋯(z−aₙ), ∀z∈𝒜 where the term with cap is missing.

The derivative is the sum of ordered products with one factor omitted. The gloss in Conjecture CSMALE (Conjecture 2.1, p. 4) is “If z∈𝒜 is not a critical point of P (i.e., P′(z)≠0)”.

Section 3, p. 7, states Conjecture DUALSMALE (Conjecture 3.1):

C-algebraic Dubinin-Sugawa Dual Mean Value Conjecture.* Let 𝒜 be a commutative C*-algebra. Let P(z) ≔ (z−a₁)⋯(z−aₙ) be a polynomial of degree n ≥ 2 over 𝒜, a₁, …, aₙ ∈ 𝒜. If z∈𝒜 is not a critical point of P, then there exists a critical point w∈𝒜 of P such that ‖P′(z)‖/deg(P) = ‖P′(z)‖/n ≤ ‖P(z)−P(w)‖/‖z−w‖.

The same norm-form assertion is Conjecture 3.1 in the journal version. Theorem 3.3 of the arXiv version (p. 7), numbered Theorem 3.1 in the journal, states:

Conjecture 3.1 holds for degree 2 C*-algebraic polynomials.

For degree two the sole critical point is (a₁+a₂)/2 and the norm inequality is an equality. This result is distinct from the degree-three refutation.

The Lean encoding uses zero-based Fin n indices and the ordered product (List.ofFn fun i => z - a i).prod. It reuses the ordered derivative of CStarSchoenberg. Noncritical means that derivative is nonzero; critical means that it is zero. The degree parameter n is cast to ℝ in the norm inequality. The carrier is restricted to unital commutative C*-algebras in Type, weakening the source’s universal assertion; a counterexample in that subclass refutes the source statement.