How many limit cycles can a planar polynomial vector field have when only the number of monomials is fixed, with the degree left arbitrary? Gasull and Santana introduced the Hilbert monomial number HM(m)\mathcal{H}^M(m) — the supremum of the number of limit cycles over all planar polynomial vector fields whose two components contain altogether exactly mm monomials — and proved quadratic-in-mm lower bounds, but no upper bound of any kind is known, and it is open whether HM(4)\mathcal{H}^M(4) is finite. The problem asks for (a) finiteness or a nontrivial upper bound for HM(m)\mathcal{H}^M(m) as a function of mm (or an explicit fixed-monomial family with unboundedly many limit cycles proving infinitude), (b) sharper small-mm values, in particular a 5- or 6-monomial field with more than 12 limit cycles or a proof of optimality, and (c) the source's Newton-polygon variation: a bound on the number of distinct periodic annuli of a Hamiltonian polynomial field in terms of the interior lattice points of its Newton polygon. As of 2026, no published work resolves any of the three components.

Background

The second part of Hilbert's 16th problem asks for the maximum number of limit cycles — isolated periodic orbits of the flow — of a planar polynomial vector field of a given degree, together with their possible configurations. Writing Xn\mathcal{X}_n for the family of planar polynomial fields X=(P,Q)X=(P,Q) with max⁡(deg⁡P,deg⁡Q)=n\max(\deg P,\deg Q)=n and π(X)\pi(X) for the number of limit cycles of XX (the value ∞\infty being admitted), the Hilbert number is H(n)=sup⁡{π(X):X∈Xn}\mathcal{H}(n)=\sup\{\pi(X):X\in\mathcal{X}_n\}. No upper bound on H(n)\mathcal{H}(n) is known for any n≥2n\geq 2 — it is not even known whether H(2)\mathcal{H}(2) is finite — while the best lower bounds grow at least on the order of n2log⁡nn^2\log n (Christopher and Lloyd 1995; Han and Li 2012), and specific examples give H(2)≥4\mathcal{H}(2)\geq 4, H(3)≥13\mathcal{H}(3)\geq 13 (Li, Liu and Yang 2009) and H(4)≥28\mathcal{H}(4)\geq 28 (Prohens and Torregrosa 2019).

Gasull and Santana (2024) proposed a sparsity-based variant, motivated by Descartes' rule of signs: a univariate polynomial with mm nonzero monomials has at most m−1m-1 positive real roots regardless of its degree. Let Mm\mathcal{M}_m be the family of planar polynomial vector fields whose two components together contain exactly mm nonzero monomials, with no restriction on the degree, and define the Hilbert monomial number HM(m)=sup⁡{π(X):X∈Mm}∈Z≥0∪{∞}\mathcal{H}^M(m)=\sup\{\pi(X):X\in\mathcal{M}_m\}\in\mathbb{Z}_{\geq 0}\cup\{\infty\}. The real setting is essential: in complex coordinates z˙=F(z,zˉ)\dot z=F(z,\bar z) with FF a polynomial with mm monomials, one and two monomials allow at most 0 and 1 limit cycles, but three monomials already allow unboundedly many (Gasull, Li and Torregrosa 2015), so no monomial-count bound can hold in the complex setting. In the real setting, Buzzi, Carvalho and Gasull (2021) proved HM(m)=0\mathcal{H}^M(m)=0 for m∈{1,2,3}m\in\{1,2,3\} and HM(m)≥m−3\mathcal{H}^M(m)\geq m-3 for m≥4m\geq 4. Gasull and Santana (2024) proved HM(m)≥12m2−3m−8\mathcal{H}^M(m)\geq\frac{1}{2}m^2-3m-8 for m≥9m\geq 9, via Abelian integrals (integrals of polynomial one-forms over the closed ovals of a Hamiltonian level set, whose zeros control limit cycles bifurcating from them), and the small-mm bounds HM(m)≥12\mathcal{H}^M(m)\geq 12 for m∈{4,5,6}m\in\{4,5,6\}, HM(7)≥16\mathcal{H}^M(7)\geq 16, HM(8)≥20\mathcal{H}^M(8)\geq 20, HM(9)≥24\mathcal{H}^M(9)\geq 24, HM(10)≥32\mathcal{H}^M(10)\geq 32; the m=9m=9 bound incorporates the computer-assisted proof of Bréhard, Brisebarre, Joldes and Tucker (2024) that a quartic field with nine monomials has at least 24 limit cycles. The m=5,6m=5,6 values are the m=4m=4 example propagated by persistence of odd-multiplicity cycles, and the authors state they believe these bounds could be improved but were unable to do so. Crucially, the paper establishes no upper bound of any kind: finiteness of HM(m)\mathcal{H}^M(m) is open even for m=4m=4, despite M4\mathcal{M}_4 containing fields of arbitrarily high degree.

The same paper's closing section, crediting its referees, proposes a geometric companion question. For a polynomial, its Newton polygon is the convex hull, in the exponent plane R2\mathbb{R}^2, of the exponent vectors (i,j)(i,j) of its monomials xiyjx^iy^j; for a polynomial vector field X=(P,Q)X=(P,Q), N(X)N(X) denotes the convex hull of the union of the exponent vectors occurring in the two components PP and QQ. In real algebraic geometry, the classical Harnack bound on the number of connected components of a real algebraic plane curve admits a sparse analogue depending only on the number of integer lattice points in the interior of the Newton polygon. A Hamiltonian vector field XH=(∂H/∂y,−∂H/∂x)X_H=(\partial H/\partial y,-\partial H/\partial x) has the level curves of the polynomial HH as trajectories; around a center — an equilibrium surrounded only by periodic orbits — its periodic orbits fill a period annulus, a maximal open annulus foliated by periodic orbits. For a Hamiltonian field, differentiation shifts each exponent vector of HH by (0,−1)(0,-1) in ∂H/∂y\partial H/\partial y and by (−1,0)(-1,0) in −∂H/∂x-\partial H/\partial x, so N(XH)N(X_H) is determined by the support of HH up to these bounded shifts; this suggests transferring the sparse algebraic-geometry bound to dynamics: bounding the number of distinct periodic annuli of XHX_H in terms of the number of interior lattice points of N(XH)N(X_H). No result in either direction is known.

Problem Statement

Determine to what extent discrete sparsity data of a planar polynomial vector field controls its periodic-orbit structure. The problem comprises three linked targets that jointly serve this one objective.

(a) Monomial Hilbert number. With Mm\mathcal{M}_m the family of planar polynomial vector fields whose two components together contain exactly mm nonzero monomials (degree arbitrary) and HM(m)=sup⁡{π(X):X∈Mm}∈Z≥0∪{∞}\mathcal{H}^M(m)=\sup\{\pi(X):X\in\mathcal{M}_m\}\in\mathbb{Z}_{\geq 0}\cup\{\infty\}, where π(X)\pi(X) is the number of limit cycles (isolated periodic orbits) of XX: determine whether HM(m)\mathcal{H}^M(m) is finite for each fixed m≥4m\geq 4 — in particular for m=4m=4. Either establish a nontrivial upper bound, i.e. an explicit function ff with HM(m)≤f(m)\mathcal{H}^M(m)\leq f(m) for all mm; or prove that HM(m)\mathcal{H}^M(m) is finite for some fixed m≥4m\geq 4 (in particular for m=4m=4); or prove that HM(m0)=∞\mathcal{H}^M(m_0)=\infty for some finite m0m_0 by exhibiting a family of vector fields with a fixed number m0m_0 of monomials and unboundedly many limit cycles. Each of these three alternatives resolves (a).

(b) Small-mm sharpness. Construct an explicit planar polynomial vector field with exactly 5 or exactly 6 monomials and more than 12 limit cycles, or prove that HM(5)=12\mathcal{H}^M(5)=12 or HM(6)=12\mathcal{H}^M(6)=12 (the current lower bounds, obtained by persistence from the 4-monomial example, are believed improvable by the source authors, who state they could not improve them).

(c) Newton-polygon annuli bound. For a Hamiltonian planar polynomial vector field XX with Newton polygon N(X)N(X), prove or refute the existence of a bound, depending only on the number of integer lattice points in the interior of N(X)N(X), on the number of distinct periodic annuli of XX. Refutation may take the form of an explicit sequence of Hamiltonian fields whose Newton polygons have a fixed number of interior lattice points while the number of distinct periodic annuli grows without bound.

The three targets share one scientific objective — a Descartes-type theory bounding the number of limit cycles and periodic annuli of planar polynomial fields by combinatorial data of their monomial support. The source presents (a) as the central open question of its variant (its own theorems are lower bounds only), (b) as an explicitly identified improvable direction, and (c) as a variation suggested by the paper's referees. The three targets are independent: resolving any one of (a), (b), (c) is a partial resolution of the problem, and resolving all three resolves it.

References

  1. Armengol Gasull, Paulo Santana, On a variant of Hilbert's 16th problem, Nonlinearity 37 (2024), no. 12, 125012. DOI: 10.1088/1361-6544/ad8c1b; arXiv:2405.04281; https://arxiv.org/abs/2405.04281
  2. Claudio A. Buzzi, Yagor Romano Carvalho, Armengol Gasull, Limit cycles for some families of smooth and non-smooth planar systems, Nonlinear Analysis 207 (2021) 112298. DOI: 10.1016/j.na.2021.112298; arXiv:2008.07692; https://arxiv.org/abs/2008.07692
  3. Armengol Gasull, Some open problems in low dimensional dynamical systems, SeMA Journal 78 (2021) 233–269. DOI: 10.1007/s40324-021-00244-3; arXiv:2012.02524; https://arxiv.org/abs/2012.02524
  4. A. Gasull, C. Li, J. Torregrosa, Limit cycles for 3-monomial differential equations, Journal of Mathematical Analysis and Applications 428 (2015) 735–749. DOI: 10.1016/j.jmaa.2015.03.039; https://doi.org/10.1016/j.jmaa.2015.03.039
  5. M. Han, J. Li, Lower bounds for the Hilbert number of polynomial systems, Journal of Differential Equations 252 (2012) 3278–3304. DOI: 10.1016/j.jde.2011.11.024; https://doi.org/10.1016/j.jde.2011.11.024
  6. Colin Christopher, Nigel G. Lloyd, Polynomial systems: a lower bound for the Hilbert numbers, Proceedings of the Royal Society of London Series A 450 (1995) 219–224. DOI: 10.1098/rspa.1995.0081; https://doi.org/10.1098/rspa.1995.0081
  7. C. Li, C. Liu, J. Yang, A cubic system with thirteen limit cycles, Journal of Differential Equations 246 (2009) 3609–3619. DOI: 10.1016/j.jde.2009.01.038; https://doi.org/10.1016/j.jde.2009.01.038
  8. R. Prohens, J. Torregrosa, New lower bounds for the Hilbert numbers using reversible centers, Nonlinearity 32 (2019) 331–355. DOI: 10.1088/1361-6544/aae94d; https://doi.org/10.1088/1361-6544/aae94d
  9. F. Bréhard, N. Brisebarre, M. Joldes, W. Tucker, Efficient and Validated Numerical Evaluation of Abelian Integrals, ACM Transactions on Mathematical Software 50 (2024) 1–38. DOI: 10.1145/3637550; https://doi.org/10.1145/3637550
  10. Jianfeng Huang, Renhao Tian, Yulin Zhao, A Chebyshev criterion for at most two non-zero limit cycles in Abel equations, Nonlinearity 39 (2026) 015030. DOI: 10.1088/1361-6544/ae3b8f; arXiv:2506.14091; https://arxiv.org/abs/2506.14091
  11. L. Q. Arakaki, D. Novaes, Strict increase in the number of normally hyperbolic limit tori in 3D polynomial vector fields, preprint (2025). arXiv:2504.13832; https://arxiv.org/abs/2504.13832
  12. M. J. Álvarez, B. Coll, A. Gasull, R. Prohens, More limit cycles for complex differential equations with three monomials, Journal of Differential Equations 416 (2025) 1071–1098. DOI: 10.1016/j.jde.2024.10.013; https://doi.org/10.1016/j.jde.2024.10.013
  13. Armengol Gasull, Paulo Santana, A note on Hilbert 16th problem, Proceedings of the American Mathematical Society 153 (2025) 669–677. DOI: 10.1090/proc/17116; arXiv:2407.13465; https://arxiv.org/abs/2407.13465