Number of centers

Definitions and mathematical statement

Let $X=(P,Q)$ be a real planar polynomial vector field of degree at most $n$, meaning $\max(\deg P,\deg Q)\le n$. An equilibrium $z_0$ satisfies $P(z_0)=Q(z_0)=0$. It is a center if it is isolated and has a punctured neighborhood filled by nonconstant periodic orbits surrounding it. This definition includes degenerate centers: no nonvanishing determinant or prescribed pair of eigenvalues is assumed for $DX(z_0)$.

Define \[ \mathcal C_n=\sup_{\deg X\le n} \#\{z_0\in\mathbb R^2:z_0\text{ is a center of }X\}. \] The problem is to determine $\mathcal C_n$ for every integer $n\ge4$. A complete determination must give both an upper bound for all real coefficient choices and examples attaining that bound, or clarify whether the supremum is finite and attained. Distinct centers are counted by their locations, not by the number of periodic orbits in their surrounding annuli.

Common polynomial factors in $P$ and $Q$ are allowed, but non-isolated equilibrium curves are not counted as centers. Counting only nondegenerate linear centers would change the problem. Nor is the task the classification of centers at one preassigned point: several different center regions can coexist in a single vector field. The total degree restriction, rather than separate bounds on each individual exponent, is the one used in defining the sequence $\mathcal C_n$.

Short English statement

Find the greatest number of distinct centers that one degree-$n$ planar polynomial vector field can have, for $n\ge4$.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 4700010 (AMR-046-0010), 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 10: Number of centers. Original source indexed by AMR-046-0010. https://arxiv.org/abs/2012.02524.