A limit cycle of a planar polynomial vector field is the image of a
nonconstant periodic solution that is isolated in some open neighbourhood.
The vector field is (P(x,y), Q(x,y)) with real polynomial coefficients,
and its degree is the larger of the total degrees of P and Q. Counting
limit cycles as the number of such images is the same as counting closed
orbits without multiplicity.

The question how many limit cycles a planar polynomial vector field can
have was posed by Hilbert in 1900 in his Paris address, as the second half
of the sixteenth of his twenty-three problems. Hilbert asked explicitly
for the maximum number of limit cycles of a planar polynomial vector field
of a given degree, and for their relative positions.

https://people.reed.edu/~davidp/341/resources/hilbert.pdf

The question splits into two clearly different assertions. The first is
individual finiteness: any fixed polynomial vector field in the plane has
finitely many limit cycles. The second is uniform boundedness: there is a
function B(d), depending only on the degree and not on coefficients or
positions, bounding the number of limit cycles of every planar polynomial
vector field of degree at most d. Individual finiteness does not imply
uniform boundedness; a family of fields can each have finitely many cycles
whose number grows without bound.

Individual finiteness was argued by Henri Dulac in a 1923 memoir in the
Bulletin de la Societe Mathematique de France. Dulac's method used a
transserial asymptotic expansion of the holonomy map attached to a nested
collection of polycycles, and concluded by a finiteness principle for
iterated integrate-exponential hierarchies. The argument was accepted for
many decades but parts of it, especially the nonaccumulation step, were
later found to need substantial repair.

https://www.numdam.org/articles/10.24033/bsmf.1031/

Complete proofs were published independently by Yulij Ilyashenko in the
1980s and by Jean Ecalle in 1992. Ilyashenko's Russian Mathematical
Surveys articles of 1985 and 1990 were followed by his monograph in the
American Mathematical Society Translations of Mathematical Monographs in
1991. Ecalle's Introduction aux fonctions analysables et preuve
constructive de la conjecture de Dulac (Hermann, 1992) introduced
resurgent analysable functions and gave a constructive version of the
proof.

https://www.mathnet.ru/eng/rm2786
https://www.mathnet.ru/eng/rm4718
https://www.ams.org/books/mmono/094/
https://www.imo.universite-paris-saclay.fr/~jean.ecalle/ecadulac.html

Ilyashenko's 2002 survey "Centennial history of Hilbert's sixteenth
problem" in the Bulletin of the American Mathematical Society traces the
resulting state of the problem. The individual finiteness result is
regarded as settled, and its methods -- monodromy, polycycle holonomy,
resurgent expansions -- form the core of subsequent work. Uniform
boundedness remains a separate, much harder assertion.

https://doi.org/10.1090/S0273-0979-02-00946-1

For degree one, uniform boundedness is trivial: linear planar systems have
no limit cycles, because closed orbits, when they exist at all, are not
isolated. For degree two the situation is already open. The best explicit
lower bound is four limit cycles, constructed by Shi Songling in 1980
(Scientia Sinica, Series A, 23, 153--158), and no uniform upper bound is
known. The best known local statement is Bautin's theorem of 1952: a
quadratic vector field with a nondegenerate centre or focus has at most
three limit cycles bifurcating from that singular point under a
one-parameter perturbation, and three is attained.

https://www.mathnet.ru/eng/sm5421

Stephen Smale's 1998 "Mathematical problems for the next century" list
in the Notices of the American Mathematical Society singled out the same
uniformity question for the quadratic case, which records how far the
state of the field has moved past Hilbert's own formulation: even the
simplest non-linear degree remains open.

https://www.ams.org/notices/199802/smale.pdf

The partial results mostly concern local estimates around a polycycle.
Ilyashenko and Yakovenko's contribution "Finite cyclicity of elementary
polycycles in generic families" (AMS Translations, Series 2, Volume 165,
1995) establishes local finite cyclicity of elementary polycycles in
generic families of finite-jet unfoldings. Vadim Kaloshin's 2003
Inventiones paper quantifies this in the existential version of the
sixteenth problem. Robert Roussarie's 1998 monograph Bifurcation of
Planar Vector Fields and Hilbert's Sixteenth Problem organises the
local-to-global reduction around the return-map graph and remains the
standard reference.

https://doi.org/10.1090/trans2/165/02
https://doi.org/10.1007/s00222-002-0244-9
https://doi.org/10.1007/978-3-0348-8798-4

The infinitesimal Hilbert sixteenth problem asks for a bound on the number
of zeros of Abelian integrals of Hamiltonian perturbations of degree d,
which is weaker than bounding the number of limit cycles. This version
was settled by Binyamini, Novikov, and Yakovenko in 2010, with explicit
bounds vastly larger than the cyclicity they govern. That result gives
the constructive side of the theory a solid footing but does not address
the uniform bound that Hilbert actually asked about.

https://doi.org/10.1007/s00222-010-0244-0

A parallel strand passes through o-minimality, model theory, and
preparation theorems. Lion and Rolin proved a preparation theorem for
logarithmico-exponential functions in 1997. Bierstone and Milman's 1988
survey Semianalytic and subanalytic sets gave the analytic foundations.
Kaiser, Rolin, and Speissegger proved in 2009 that the transition map at
a non-resonant hyperbolic singularity is definable in an o-minimal
expansion; Galal, Kaiser, and Speissegger extended this to Ilyashenko
algebras with injective transserial asymptotics in 2020. These give
definability and uniformity for restricted local classes, not a global
uniform bound.

https://www.numdam.org/item/10.5802/aif.1583.pdf
https://www.numdam.org/item/10.1007/BF02699126.pdf
https://doi.org/10.1515/CRELLE.2009.081
https://doi.org/10.1016/j.aim.2020.107095

A recent revisiting of Dulac's classical argument is Yeung's "Dulac's
Theorem Revisited" in Qualitative Theory of Dynamical Systems (2025),
which re-examines the differentiable closure steps of the original proof.
Such work consolidates the individual-finiteness side; it does not settle
uniform boundedness.

https://doi.org/10.1007/s12346-025-01220-2

These strands do not add up to a proof of uniform boundedness in degree
two or in any higher degree. Every local passage must be made uniform in
the coefficients, in the positions of the singularities, and across all
moving polycycles; the resurgent and the o-minimal methods contribute to
that programme piece by piece. As the standing literature records, no
accepted proof of uniform boundedness for general degree exists.

Any argument that claims to settle uniform boundedness for all d is
therefore claiming to resolve a problem listed as open in the standard
references, including Smale's problem list. Such an argument would have to
be checked line by line against the local cyclicity results, the
preparation and definability estimates, and the individual-finiteness
machinery sketched above. A careful reader should compare any such
argument with Ilyashenko's 2002 survey and with Roussarie's 1998
monograph before relying on it.