A piecewise-linear Hilbert sixteenth-type problem

Definitions and mathematical statement

Let $H(n)$ be the supremum of the number of isolated nonconstant periodic orbits of a real planar polynomial vector field of total degree at most $n$. For a second class, let a smooth embedded branch $\Sigma$ of a real degree-$n$ algebraic curve separate the plane into two zones. Put an affine linear vector field in each zone, allowing different fields on the two sides. A crossing periodic orbit follows these fields and crosses $\Sigma$ transversely, with the two normal components having the same nonzero sign at every crossing. No sliding segment or tangency is included.

Let $L(n)$ be the corresponding supremum of the number of isolated crossing periodic orbits, over admissible separating branches and both affine fields. The source asks to improve the lower bounds \[ H(2)\ge4,\quad H(3)\ge13,\quad H(n)\ge K n^2\log n, \qquad L(1)\ge3,\quad L(2)\ge4,\quad L(n)\ge\lfloor n/2\rfloor, \] where the asymptotic estimate uses an absolute $K>0$ for sufficiently large $n$. These are the source's historical benchmarks, not assertions that they remain the best published estimates.

The continuing construction problem is to determine how large these quantities can be and improve available degree-dependent guarantees. Each construction must verify isolation of every counted orbit and, for $L(n)$, the prescribed two-zone geometry and transverse crossing condition. Adding more zones or allowing a nonlinear vector field inside a zone changes the class and does not establish a bound for $L(n)$.

Short English statement

Improve constructions with many limit cycles in polynomial systems and in two-zone piecewise-linear systems with algebraic separation.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 4700018 (AMR-046-0018), 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 18: A piecewise-linear Hilbert sixteenth-type problem. Original source indexed by AMR-046-0018. https://arxiv.org/abs/2012.02524.