The Bojanov–Naidenov extremal problem for the k-th derivative of algebraic polynomials: the finite cube-vertex check, settled in every degree n ≤ 6
Abstract
Let πₙᶜⁱʳᶜ be the real polynomials of degree at most n bounded by 1 on [-1,1] and Tₙ the Chebyshev polynomial. Bojanov's problem, as recorded by Naidenov, asks whether ∫₋₁¹φ(|P^((k))|)<∫₋₁¹φ(|Tₙ^((k))|) for every strictly increasing convex φ, every P ∈ πₙᶜⁱʳᶜsetminus{± Tₙ} and every 2 ≤ k ≤ n-1. Inside that range only (n,k)=(4,2) was settled, by Zavalani (arXiv:2606.23020), who interpolates at the n+1 extremal points of Tₙ, passes by convexity to the cube of nodal values, and checks the 2ⁿ⁺¹ vertices. We run that reduction in exact arithmetic over ℚ(sqrt D) at all ten pairs 3 ≤ n ≤ 6, 2 ≤ k ≤ n-1: at every vertex y ∈ {-1,1}ⁿ⁺¹ the integrated tails ∫(|P_y^((k))|-t)₊ are dominated by those of Tₙ^((k)) at every level t ≥ 0, the bound is attained at ± Tₙ, and the extremal sup norm is Tₙ^((k))(1) (24,80,192,200,840,1920,420,2688,10368,23040). Combined with the elementary convexity and tail-to-convex-order lemmas of the source, this gives the Bojanov–Naidenov inequality in its non-strict form — for every nondecreasing convex φ — at the nine previously open pairs, and under the weaker Duffin–Schaeffer hypothesis |P(cos(jπ/n))| ≤ 1, j=0,…,n; we point out that the sup-norm content of every such statement is Duffin and Schaeffer's Theorem III of 1941, which the source proves at (4,2), under the stronger hypothesis norm(P)_(C[-1,1]) ≤ 1, without citing it. A second finding concerns the method: the pointwise level-set domination through which the source proves its vertex bound is false at nine of the ten pairs, (4,2) being the only one where it holds — certified at six pairs by explicit vertex witnesses whose k-th derivative stays away from zero on [-1,1], and found by external computation at the other three. Each vertex is certified by a transport between two uniform partitions with degree-four Bernstein enclosures, whose only data per vertex is a pair of partition sizes; the certificates, the setup and the witnesses are verified in Lean 4 against Mathlib.
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Claim ledger
Stated results
BN1candidate2026-09-07
Bojanov-Naidenov vertex check at (n,k) = (3,2): for every sign vector y in -1,1⁴, the integrated tails of the 2-th derivative of the degree-3 interpolant of y at the extremal points of T₃ are dominated by those of T₃² at every level t >= 0
BN2known2026-09-07
Bojanov-Naidenov vertex check at (n,k) = (4,2): for every sign vector y in -1,1⁵, the integrated tails of the 2-th derivative of the degree-4 interpolant of y at the extremal points of T₄ are dominated by those of T₄² at every level t >= 0
BN3candidate2026-09-07
Bojanov-Naidenov vertex check at (n,k) = (4,3): for every sign vector y in -1,1⁵, the integrated tails of the 3-th derivative of the degree-4 interpolant of y at the extremal points of T₄ are dominated by those of T₄³ at every level t >= 0
BN4candidate2026-09-07
Bojanov-Naidenov vertex check at (n,k) = (5,2): for every sign vector y in -1,1⁶, the integrated tails of the 2-th derivative of the degree-5 interpolant of y at the extremal points of T₅ are dominated by those of T₅² at every level t >= 0
BN5candidate2026-09-07
Bojanov-Naidenov vertex check at (n,k) = (5,3): for every sign vector y in -1,1⁶, the integrated tails of the 3-th derivative of the degree-5 interpolant of y at the extremal points of T₅ are dominated by those of T₅³ at every level t >= 0
BN6candidate2026-09-07
Bojanov-Naidenov vertex check at (n,k) = (5,4): for every sign vector y in -1,1⁶, the integrated tails of the 4-th derivative of the degree-5 interpolant of y at the extremal points of T₅ are dominated by those of T₅⁴ at every level t >= 0
BN7candidate2026-09-07
Bojanov-Naidenov vertex check at (n,k) = (6,2): for every sign vector y in -1,1⁷, the integrated tails of the 2-th derivative of the degree-6 interpolant of y at the extremal points of T₆ are dominated by those of T₆² at every level t >= 0
BN8candidate2026-09-07
Bojanov-Naidenov vertex check at (n,k) = (6,3): for every sign vector y in -1,1⁷, the integrated tails of the 3-th derivative of the degree-6 interpolant of y at the extremal points of T₆ are dominated by those of T₆³ at every level t >= 0
BN9candidate2026-09-07
Bojanov-Naidenov vertex check at (n,k) = (6,4): for every sign vector y in -1,1⁷, the integrated tails of the 4-th derivative of the degree-6 interpolant of y at the extremal points of T₆ are dominated by those of T₆⁴ at every level t >= 0
BN10candidate2026-09-07
Bojanov-Naidenov vertex check at (n,k) = (6,5): for every sign vector y in -1,1⁷, the integrated tails of the 5-th derivative of the degree-6 interpolant of y at the extremal points of T₆ are dominated by those of T₆⁵ at every level t >= 0
BN11candidate2026-09-07
The pointwise level-set route of arXiv:2606.23020v1 (its Lemma 6, equation (6.1): meas|P_yᵏ| > t <= meas|Tₙᵏ| > t at every cube vertex) FAILS at nine of the ten cells with n <= 6 – every cell except the source's own (4,2) – with explicit kernel-bound cube-vertex witnesses at six of them; only the weaker integrated-tail comparison (6.2) survives
BN12routine2026-09-07
Soundness of the two-partition Bernstein transport certificate: matching cells of two uniform partitions of [-1,1] with certified sup and inf bounds, so that each A-cell consumes a block of B-cells of at least its own height and at least its own area, proves domination of the integrated tails at every level t >= 0
BN13routine2026-09-07
Negative controls at all ten cells: the right-hand side cannot be shrunk by any factor lambda < 1 (the Chebyshev cube vertex already violates it), and the non-Chebyshev cube vertex of largest L¹ norm is strictly below Tₙᵏ at t = 0
BN14known data2026-09-07
The setup, kernel-checked at all ten cells: the nodes are the n+1 extremal points of Mathlib's Polynomial.Chebyshev.T R n in decreasing order with Tₙ(xⱼ) = (-1)ʲ, the Lagrange basis satisfies lⱼ(xᵢ) = deltaᵢj, each cube vertex interpolates its sign vector, and Tₙᵏ is computed explicitly
BN15measurement2026-09-07
Cost of the ten-cell check: about 35 CPU-min end to end, maximum partition sizes N_A = 128 and N_B = 684 (both at (6,2)), peak resident set about 3 GB with one Lean process at a time
This ledger entry is reported in prose and is not bound to a Lean theorem.Provenance
- Generated by
- Machina Mathematica
- Released by
- Korea Superintelligence Labs
- Source context
- Let πₙ be the real polynomials of degree at most n, and
- Snapshot
- 2026-09-07 03:53 UTC
- Ledger commit
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