# P9: Hilbert's sixteenth problem, second part

- ID: `P9`
- Reference: `hilberts-sixteenth-problem-second-part`
- Page: https://theoremdb.org/statements/P9
- Record maturity: Reviewed problem with recorded work

## Problem

For every positive integer \(n\), there exists a finite number \(H(n)\) such that each planar polynomial vector field of degree at most \(n\) has at most \(H(n)\) limit cycles.

### Context

The problem asks for degree-controlled global information about isolated periodic behavior in polynomial differential systems.

### Problem setup

- **Definition (A limit cycle).** A limit cycle is an isolated periodic orbit of a planar differential equation.
- **Definition (The bound may depend on the polynomial degree n but must apply uniformly to every vector field of that degree).** The bound may depend on the polynomial degree n but must apply uniformly to every vector field of that degree.
- **Remark.** The problem asks for degree-controlled global information about isolated periodic behavior in polynomial differential systems.

### What counts as a solution

- Prove a finite uniform bound H(n) for every degree n, or construct a fixed degree with polynomial vector fields having arbitrarily many limit cycles.

## Status

Unresolved in this packet after the dated source check. Strongest checked result: Buzzi and Novaes refute a recent claimed quadratic formula by comparing it with established lower-growth constructions. Their note leaves the finiteness of H(n) open. Exact unresolved remainder: Prove a finite uniform upper bound H(n) for every degree n, or give one fixed degree with polynomial vector fields having arbitrarily many limit cycles. [1](#reference-1)

## Work

### Evidence for the current status

**Claim 1 (Dated status and exact unresolved remainder).** Unresolved in this packet after the dated source check. Strongest checked result: Buzzi and Novaes refute a recent claimed quadratic formula by comparing it with established lower-growth constructions. Their note leaves the finiteness of H(n) open. Exact unresolved remainder: Prove a finite uniform upper bound H(n) for every degree n, or give one fixed degree with polynomial vector fields having arbitrarily many limit cycles.

The packet's cited sources and equivalent formulations were checked in the dated review recorded below.

Strongest checked result: Buzzi and Novaes refute a recent claimed quadratic formula by comparing it with established lower-growth constructions. Their note leaves the finiteness of H(n) open.

Exact unresolved remainder: Prove a finite uniform upper bound H(n) for every degree n, or give one fixed degree with polynomial vector fields having arbitrarily many limit cycles.

### Background and intake notes

- Original intake status: The cited 2024 note treats the existence of the uniform finite bounds H(n) as unresolved. The source and public status were checked on 2026-07-22. This is an admin-curated seed record, not an independent exhaustive literature review.
- The standard finiteness formulation was checked against the cited note on 2026-07-22.
- Finiteness for each individual polynomial vector field is known. Uniform finiteness by degree remains open even in low degrees.

### Open directions

- **Route 1** (reported): Prove a finite uniform bound H(n) for every degree n, or construct a fixed degree with polynomial vector fields having arbitrarily many limit cycles. [1](#reference-1)

### Computational notes

- Numerical integration can locate cycles in selected systems but can miss cycles and cannot establish a universal degree bound.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `hilberts-sixteenth-problem-second-part`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>Claudio A. Buzzi and Douglas D. Novaes, “A note on a recent attempt to solve the second part of Hilbert's 16th Problem”. arXiv:2411.09594 (2024). arXiv:2411.09594, opening formulation and analysis of a claimed solution https://arxiv.org/abs/2411.09594
   - Also cited at abstract and analysis of the claimed quadratic formula
   - Also cited at Editorial research route recorded 2026-07-31
   - preprint; primary source; arXiv:2411.09594, checked 2026-07-31; checked 2026-07-31
   - Source use: original_summary
   - The cited 2024 note treats the existence of the uniform finite bounds H(n) as unresolved. The source and public status were checked on 2026-07-22. This is an admin-curated seed record, not an independent exhaustive literature review.
   - Source used to formulate or check the problem record.
   - Source used to assess the problem's recorded status.
   - Rejects a claimed solution while preserving the exact finiteness question.
   - Source named by the research packet.