Nikolov Snake Polynomial Chebyshev Expansion Refutation
Abstract
An even convex majorant has a degree-six snake with a negative Chebyshev coefficient.
Kryvonos (ed.), arXiv:2312.07754v1, section “Snake polynomials with positive Chebyshev expansions”, definition verbatim: “Consider a majorant µ ∈ C[−1, 1], µ(x) ≥ 0, x ∈ [−1, 1]. If there exists P ∈ Pn, P ≠ 0, such that −µ(x) ≤ P(x) ≤ µ(x), x ∈ [−1, 1], then there exists a unique (up to orientation) polynomial ωµ ∈ Pn called snake polynomial associated with µ, which oscillates most between ±µ.” “The n-th snake polynomial ωµ is uniquely determined by the following properties: a) |ωµ(x)| ≤ µ(x) x ∈ [−1, 1]; b) There exists a set δ* = (τi*)_{i=0}^n, 1 ≥ τ0* > · · · > τn* ≥ −1, such that ωµ(τi*) = (−1)^i µ(τi*), i = 0, . . . , n” Conjecture verbatim: “Conjecture (G.N.) If µ ≥ 0 is a continuous even convex function in [−1, 1], then the associated with µ snake polynomials have non-negative expansion in the Chebyshev polynomials of the first kind.”
Definition 1.1 (Bound and alternating contacts).
Formalization. D5/S3/Analytic/Chebyshev/SnakeChebyshevExpansionRefutation.IsSnake (✓ std3).
Citation. L. Kryvonos (ed.) (2023). Open problems on polynomials, their zero distribution and related questions: 2023. URL: https://arxiv.org/abs/2312.07754v1.
Commentary.
The formal encoding of the quoted characterization uses a real polynomial of degree at most n, a pointwise bound on I=[−1,1], and a strictly decreasing map τ from Fin(n+1) to ℝ. Its first contact is at most one, its last at least minus one, and its values alternate as ω(τᵢ)=(−1)ⁱμ(τᵢ). These conditions express (a) and (b) in the positive initial orientation. Uniqueness up to orientation is asserted by the source and is not formalized here.
Definition 1.2 (Nonnegative Chebyshev expansion).
Formalization. D5/S3/Analytic/Chebyshev/SnakeChebyshevExpansionRefutation.claim (✓ std3).
Citation. L. Kryvonos (ed.) (2023). Open problems on polynomials, their zero distribution and related questions: 2023. URL: https://arxiv.org/abs/2312.07754v1.
Commentary.
The formal claim quantifies over every real function continuous, nonnegative, even and convex on I, every natural n and every polynomial satisfying IsSnake. It requires a coefficient function a:ℕ→ℝ nonnegative at every index, with ω=∑ₖ₌₀ⁿ C(aₖ)Tₖ. Tₖ is Chebyshev.T ℝ (k:ℤ), the first-kind polynomial, and C embeds a real constant. UP24, arXiv:2504.04845, §7, repeats the assertion as Conjecture 7.1 (Geno Nikolov).
Theorem 1.3 (A negative constant Chebyshev coefficient).
Proof. Machine-checked in Lean as D5/S3/Analytic/Chebyshev/SnakeChebyshevExpansionRefutation.result (✓ std3). ∎
Resolves. Problems/nikolov-2023-snake-polynomial-chebyshev-expansion (refuted) by D5/S3/Analytic/Chebyshev/SnakeChebyshevExpansionRefutation.result.
Source. Repository-derived.
Acknowledgement. L. Kryvonos (ed.) (2023). Open problems on polynomials, their zero distribution and related questions: 2023. URL: https://arxiv.org/abs/2312.07754v1.
Commentary.
Take n=6, μ(x)=max{1,(4860|x|−1078)/137}, and P(x)=(38656x⁶−41568x⁴+6831x²−137)/137. The majorant is positive, continuous, even and convex. On [0,1/4], factorizations of 137(1−P) and 137(1+P) prove |P|≤1. On [1/4,1], with L(x)=(4860x−1078)/137, the factors of 137(L−P) are (1−x)(4x−1)(9664x⁴+12080x³+2292x²−155x+941), and 137(L+P)=(4x−3)²(2416x⁴+3624x³+1479x²+180x−135). The last quartic equals 137/2+(x−1/4)(2416x³+4228x²+2536x+814). The interval factors are nonnegative; evenness extends the bound to [−1,1]. The contacts 1,3/4,1/4,0,−1/4,−3/4,−1 have values 3782/137,−2567/137,1,−1,1,−2567/137,3782/137, respectively. The private lemmas P_bound_small, P_bound_large and P_snake establish these snake conditions. Comparing the coefficients of x⁰,x²,x⁴,x⁶ in a representation by T₀,…,T₆ forces a₀=−459/274, as proved by constant_coefficient_negative. The exact polynomial expansion is P=(−459/274)T₀+(1503/274)T₂+(2052/137)T₄+(1208/137)T₆. Thus nonnegative coefficients are impossible. Nikolov–Shadrin, arXiv:1210.7795, uses positive expansion as a sufficient hypothesis for extremality in majorant inequalities. This refutation leaves the folklore Markov-extremality conjecture and UP24 Problem 7.1 open; it also leaves strictly convex or smoother majorants and degrees at most five unsettled. Literature coverage is limited to the cited arXiv versions.