A new Hilbert sixteenth-type problem

Definitions and mathematical statement

A monomial vector field on the plane is either $(x^ay^b,0)$ or $(0,x^ay^b)$, where $a,b$ are nonnegative integers. A nonzero scalar coefficient times each such term is counted once; identical powers in different components are distinct terms. Let $\mathcal M_m$ consist of polynomial vector fields obtained using at most $m$ distinct terms, with no restriction on their degrees. Put \[ H^M(m)=\sup_{X\in\mathcal M_m}\#\{\text{limit cycles of }X\}. \] A limit cycle is an isolated nonconstant periodic orbit, counted once regardless of its parametrization or multiplicity. The value $+\infty$ is allowed in this supremum.

The main problem is to find useful upper and lower bounds on $H^M(m)$ and, where possible, determine its exact values or growth. The original source also asks for the least number of monomials allowing at least $m+1$ cycles. That latter question is now resolved: later work gives no cycles for $m\le3$ and at least twelve for $m=4$, so its least value is four.

The broader bounds question remains the target of this definition. In particular a degree-dependent estimate does not answer it, because arbitrarily large exponents are permitted while the number of terms stays fixed. The 2024 quadratic-growth lower bound is progress on this question, rather than an upper bound or an exact determination of $H^M(m)$.

Short English statement

Bound the number of planar limit cycles by the number of monomial terms, with arbitrary exponents.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 4700008 (AMR-046-0008), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Gasull - Some open problems in low dimensional dynamical systems (2020), Problem environment 8: A new Hilbert sixteenth-type problem. Original source indexed by AMR-046-0008. https://arxiv.org/abs/2012.02524.
• S3 Gasull and Santana, On a variant of Hilbert’s 16th problem (2024), Theorems 1–2; partial progress and the resolved least-term subquestion. https://arxiv.org/abs/2405.04281.