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Loewner's conjecture

Let ff be real analytic near the origin, with f(0,0)=0f(0,0)=0, and let n>1n>1. Suppose the origin is an isolated equilibrium of

x˙=2nRe⁡ ⁣(∂nf∂zˉn),y˙=2nIm⁡ ⁣(∂nf∂zˉn).\dot x=2^n\operatorname{Re}\!\left(\frac{\partial^n f}{\partial\bar z^n}\right),\qquad \dot y=2^n\operatorname{Im}\!\left(\frac{\partial^n f}{\partial\bar z^n}\right).

Prove that the index of this vector field at the origin is at most nn.

References

Primary source

Gasull - Some open problems in low dimensional dynamical systems (2020) (proposed by Armengol Gasull)

dynamical-systems
InAMR Open Problem Lists
Posted

Progress summary

Refreshed
Open

The conjecture still has no recorded proof or counterexample: it predicts an upper bound for certain isolated planar singularities, and only special cases are known.

The conjecture asserts that the index of the isolated zero in the stated planar vector field is at most nn for every n>1n>1. It is closely connected with Carathéodory's conjecture; a 2021 paper explicitly described both as open.

Known results

  • The analytic n=2n=2 case is associated with Hamburger's work from 1940–41 on Carathéodory's conjecture, but this does not settle the general statement.
  • Under negative-Gaussian-curvature hypotheses, an index non-positivity result is known; this is weaker than the conjectured upper bound.

Related developments, 2022–2024

A May 2022 preprint proves a conditional theorem giving nn interior Loewner points under strong boundary assumptions, not the unrestricted local conjecture. A February 2024 preprint studies the n=2n=2 umbilic case, but the supplied record does not claim a proof of the general conjecture.

Current status (as of August 2026): Restricted cases and conditional boundary results are known, but the stated general conjecture remains open, with no recorded proof or counterexample.

Sources

Solutions 0

No solutions have been posted yet.