The second part of Hilbert's sixteenth problem: finiteness and value of the Hilbert number $H(n)$ for planar polynomial vector fields

The second part of Hilbert's sixteenth problem (1900) asks for the maximum number of limit cycles of a planar polynomial vector field of degree $n$ and their relative positions. In its modern quantitative form the problem is to determine the Hilbert number $H(n)$, the supremum of the number of limit cycles over all real planar polynomial vector fields of degree $n$: is $H(n)$ finite, and if so what is its exact value? More than 125 years after Hilbert's address, not even the finiteness of $H(2)$ for quadratic systems is known: individual polynomial vector fields have finitely many limit cycles (Ilyashenko, Écalle), but no upper bound depending only on the degree exists in the literature, while lower bounds grow at least on the order of $n^2\log n$.

Background

A planar polynomial vector field (equivalently, polynomial system of differential equations) of degree $n$ is a pair $\dot{x}=P(x,y)$, $\dot{y}=Q(x,y)$ with $P,Q$ real polynomials of maximum degree $n$. A limit cycle is a periodic orbit of such a system that is isolated from all other periodic orbits; limit cycles model self-sustained oscillations and are, after equilibria, the simplest nontrivial compact invariant sets of planar flows. In the second part of his sixteenth problem, posed in his 1900 address to the International Congress of Mathematicians, Hilbert asked for the maximum number and the relative positions of limit cycles of a polynomial system of degree $n$. The quantitative core of the problem is captured by the Hilbert number $H(n)$, defined as the supremum of the number of limit cycles over all polynomial vector fields of degree $n$ (the supremum is over an infinite-dimensional coefficient space, so a priori $H(n)$ may be infinite; the degree convention is $\max(\deg P, \deg Q)=n$). Already for $n=2$ the finiteness of $H(2)$ is unknown: the best established lower bound is $H(2)\ge 4$, from classical quadratic examples with four limit cycles, and no quadratic system with more than four limit cycles is known. Two clarifications delimit what is and is not known. First, the individual finiteness question is settled: the assertion of Dulac (1923) that any fixed polynomial vector field has only finitely many limit cycles was proved rigorously around 1990 independently by Ilyashenko and by Écalle (via 'resurgence' and non-analytic class theories); but their arguments are non-uniform in the coefficients, so they yield no bound depending only on $n$. Second, a weakened (infinitesimal) version, bounding the number of zeros of Abelian integrals — certain integrals over ovals of level curves of a Hamiltonian polynomial that govern limit cycles born under small polynomial perturbations of Hamiltonian systems — is known to admit a universal bound depending only on the degree (Varchenko and Khovanskii, via fewnomial theory), but these restricted estimates do not transfer to the full problem. Smale, in his 1998 list of mathematical problems for the 21st century, prominently re-listed a weakened form of the question, asking whether the number of limit cycles admits an upper bound in terms of the degree alone. On the lower-bound side, Christopher and Lloyd (1995) proved that $H(n)$ grows at least on the order of $n^2\log n$ — more precisely, $H(n)\ge\tfrac{1}{2}n^2\log_2 n+O(n)$ for degrees $n$ of the form $2k-1$; Itenberg and Shustin (2000) extended this estimate to all degrees $n\ge 3$ using Viro's 'gluing' (patchworking) method for constructing real algebraic curves with prescribed topology; and Han and Li (2012) refined it to $\liminf_{n\to\infty} H(n)/((n+2)^2\log(n+2))\ge 1/(2\log 2)$, the best known asymptotic lower bound. Degree-specific lower bounds include $H(3)\ge 13$ (a cubic system with thirteen limit cycles, Li–Liu–Yang 2009) and $H(n)\ge 28,37,53,\dots$ for $n=4,5,6,\dots$ (Prohens–Torregrosa 2019, via reversible centers — a center being an equilibrium surrounded by continua of periodic orbits, here occurring in systems invariant under a symmetry that reverses the direction of time). Recently Gasull and Santana (2025) proved two structural facts about $H(n)$: it is realizable by structurally stable vector fields (fields whose orbit structure is topologically unchanged under small perturbations) all of whose limit cycles are hyperbolic (i.e., linearly isolated, with non-unit characteristic multiplier), and it satisfies $H(n+1)\ge H(n)+1$, so $H(n)$ is strictly increasing whenever finite; their companion paper (published as Communications on Pure and Applied Analysis 28 (2026), 214–220) proves the recurrent inequality $H(n+m)\ge H(n)+\mathrm{Har}(m)$ with $\mathrm{Har}(m)=(m-1)(m-2)/2+[1+(-1)^m]/2$. None of this touches the upper-bound question: no function $U(n)$ finite for all $n$ is known to satisfy $H(n)\le U(n)$.

Problem Statement

Determine the Hilbert number $H(n)$ for planar polynomial vector fields. For each integer $n\ge 2$, let $H(n)=\sup\{\pi(X): X$ a real planar polynomial vector field of degree $n\}$, where $\pi(X)$ is the number of limit cycles (isolated periodic orbits) of $X$; the supremum is taken over all coefficient choices, so $H(n)$ may a priori be infinite. Decide: (i) whether $H(n)<\infty$ — in particular whether $H(2)<\infty$, and whether there exists any finite-valued function $U(n)$ with $H(n)\le U(n)$ for all $n$; and (ii) if $H(n)$ is finite, its exact value. (Hilbert's original formulation additionally asks for the possible relative positions and configurations of the limit cycles of degree-$n$ systems; the quantitative bound $H(n)$ stated here is the standard modern core of the problem.)

References

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