Maximum number of critical periods

Definitions and mathematical statement

A planar polynomial differential system of degree at most $n$ has the form $\dot x=P(x,y)$, $\dot y=Q(x,y)$ with real polynomials of total degree at most $n$. For any center, its period annulus is the maximal surrounding family of nonconstant periodic orbits. Choosing a regular transverse coordinate $s$ on this annulus gives the least-period function $T(s)$. An isolated zero of $T'(s)$ is a critical period; the definition is independent of the chosen regular transverse coordinate.

Let $\mathcal T(n)$ be the supremum of the numbers of such critical periods on a period annulus, over all degree-at-most-$n$ systems and their centers. The problem asks whether there are constants $C>0$ and $n_0$ such that \[ \mathcal T(n)\ge C n^2\log n\qquad(n\ge n_0). \] Equivalently, can one construct, for every sufficiently large degree, a system and a center whose surrounding annulus has at least a constant multiple of $n^2\log n$ distinct stationary periods?

The logarithm is natural; changing its base only changes $C$. The same positive constant must work for all large $n$. Critical periods are counted as distinct stationary orbits, without multiplicity, and endpoints of the annulus are excluded. An isochronous annulus contributes no isolated critical periods. The intended target is an improvement over quadratic growth by a logarithmic factor, not an upper estimate on $\mathcal T(n)$ or merely a degree-dependent existence result for a single critical period.

Short English statement

Can degree-$n$ polynomial centers have at least a constant times $n^2\log n$ distinct critical periods?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 4700014 (AMR-046-0014), 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 14: Maximum number of critical periods. Original source indexed by AMR-046-0014. https://arxiv.org/abs/2012.02524.