Hilbert 16(b) — the Hilbert number H(n)

Slug: hilbert16-limit-cycles
List: Hilbert 16 (second part); Smale 13 (1998)
Solver: Grok 4.6
Status: open. No published H(n) moved. First campaign dropped H(2)≥5 and H(3)≥14. Second campaign wrote 25 new fictions and dropped the five that scored highest. Third campaign wrote ten more and dropped the official three plus two extras (two-well 14; PT L1 rank 29; qh M1 four zeros; holomorphic cube 9-sheet; beat of Z(2,n)). The leftover menu and four extras also dropped (constructive +1; cubic+line 3; five Abelian zeros; (u²,v²) or T2 beating 74; Kolmogorov 7; L2 extra wells; (x²+y²)X two cycles). Residue: every imagined dent died; certified forks live in compute/q1/ through compute/q3/. Hilbert 16(b) itself is untouched.
Area: Qualitative theory of planar polynomial ODEs
Sources: Shi, Sci. Sinica 23 (1980); Chen–Wang 1979; Bautin 1952/54; Li–Liu–Yang, JDE 246 (2009); Christopher–Lloyd, Proc. R. Soc. A 450 (1995); Han–Li, JDE 252 (2012); Prohens–Torregrosa, Nonlinearity 32 (2019); Gasull–Santana, arXiv:2407.13465; Eshkobilov–Kadyrov–Mamayusupov, arXiv:2604.12883; Coppel, JDE 2 (1966); Llibre–Schlomiuk, Canad. J. Math. 56 (2004); Chen–Dai–Kaloshin–Li, arXiv:2608.17773; Gasull–Santana, arXiv:2510.11705; Gavrilov–He–Xiao, arXiv:2606.22137; Villanueva–Tucker, arXiv:2602.22558
Started: 2026-08-27

This folder is Hilbert 16(b): isolated periodic orbits of planar polynomial vector fields. It is not the existing problems/hilbert16-degree-8/ folder, which is 16(a) for real plane octics.

In general

A planar polynomial vector field of degree n is a system

\dot x = P(x,y),\qquad \dot y = Q(x,y)

with P, Q real polynomials and max(deg P, deg Q) = n. A limit cycle is an isolated periodic orbit. The Hilbert number H(n) is the supremum of the number of limit cycles over all such fields of degree at most n (the value +∞ is allowed). Hilbert asked for a uniform finite upper bound in n and for the possible configurations. Smale restated the upper-bound half as Problem 13 and called it, after RH, the most elusive of Hilbert's problems.

Ilyashenko and Écalle independently proved that each individual polynomial field has finitely many limit cycles. That is finiteness per system, not a uniform H(n). Bamon (1986) had already proved per-system finiteness in degree 2. It is still not known whether H(2) itself is finite. Green–Smale 13 and the Ilyashenko/Écalle theorems are context, not the table.

A quadratic formula H(n) = 2(n−1)(4(n−1)−2) appeared in Entropy 26 (2024) and is not the record: it is quadratic in n, which contradicts the Christopher–Lloyd n² log n lower bound (Buzzi–Novaes, arXiv:2411.09594).

Published record (fetched 2026-08-27)

Lower bounds are constructions. arXiv and the journals below are the record; forum numbers are leads.

Small n, still cited as best in 2023–2026 papers that we opened:

H(1) = 0 (linear fields).
H(2) ≥ 4, Shi Songling, Sci. Sinica 23 (1980), and independently Chen–Wang (1979). Configuration (3,1): three small cycles from a order-3 weak focus and one large cycle. Bautin proved that a quadratic focus has cyclicity at most 3, so four small cycles at one focus are excluded.
H(3) ≥ 13, Li–Liu–Yang, J. Differ. Equations 246 (2009). Configuration (5:1|1:5) plus one surrounding cycle, via zeros of Abelian integrals. Local cyclicity M(3) ≥ 12 (Giné–Gouveia–Torregrosa, JDE 275, 2021) is weaker than 13 as a global lower bound.
H(4) ≥ 28, H(5) ≥ 37, H(6) ≥ 53, H(7) ≥ 74, H(8) ≥ 96, H(9) ≥ 120, H(10) ≥ 142, Prohens–Torregrosa, Nonlinearity 32 (2019), three-nest simultaneous Hopf from reversible Darboux centers.

Asymptotics:

Christopher–Lloyd, Proc. R. Soc. A 450 (1995): H(n) grows at least as fast as n² log n, and the four-fold step H(2n+1) ≥ 4 H(n).
Han–Li, J. Differ. Equations 252 (2012): liminf H(n) / ((n+2)² log(n+2)) ≥ 1/(2 log 2), and a table of explicit seeds (H(11) ≥ 153, H(14) ≥ 194, H(15) ≥ 345, H(19) ≥ 503, …).

Gasull–Santana, arXiv:2407.13465v2: if H(n) is finite then it is attained by a structurally stable field with only hyperbolic cycles, and H(n+1) ≥ H(n)+1.

Eshkobilov–Kadyrov–Mamayusupov, arXiv:2604.12883v1 (14 Apr 2026): separable Chebyshev pullback

H(nm+m-1)\ge m^2 H(n)\qquad(m\ge 2).

Combined with the Han–Li and Prohens–Torregrosa seeds they record H(14) ≥ 252, H(29) ≥ 1080, H(31) ≥ 1380, H(39) ≥ 2012. Those four numbers are theorem-only lifts of unreplayed seeds. This folder replays the pullback identity and the arithmetic; it does not independently reconstruct the Prohens–Torregrosa or Han–Li centers.

Restricted families (not H(n) itself): Kolmogorov local cyclicity M_K(3) ≥ 6, M_K(4) ≥ 13, M_K(5) ≥ 22 (Carvalho–Cruz–Gouveia, arXiv:2304.05111). Piecewise quadratic H_p(2) ≥ 16 is a different number.

Precise statement

Move a published H(n) lower bound by a verified construction, or prove a new exact finite upper bound for some n or for a documented restricted family, or produce a reusable exact lemma that a stranger can replay.

What would count as a new bound

A dent is a verified finite improvement of a published H(n), or a new exact upper bound. Write the inequality and the record it beats.

An incomplete search (numerical integration, SAT UNKNOWN, a Poincaré–Bendixson picture without a certificate) is residue, not a lower bound.

Replaying Shi's 4, Li–Liu–Yang's 13, or the 2026 Chebyshev arithmetic is not a new bound. A lift H(nm+m−1) ≥ m² L_pub(n) is not an independent lower bound unless the seed L_pub(n) is itself replayed here.

What does not count

Hilbert 16(a), including anything in problems/hilbert16-degree-8/.
Green–Smale 13 restated.
Ilyashenko/Écalle per-system finiteness.
Piecewise, discontinuous, or delay analogues, unless the claim is clearly labelled as a different function.
The Entropy 2024 quadratic formula.