Loewner's conjecture

Definitions and mathematical statement

Fix an integer $n>1$. Let $f$ be a real-valued real-analytic function on a neighborhood of $(0,0)$ with $f(0,0)=0$. Put $z=x+iy$ and use the Wirtinger operator \[ \partial_{\bar z}=\tfrac12(\partial_x+i\partial_y). \] Define the real planar vector field \[ X_f=2^n\left(\operatorname{Re}\partial_{\bar z}^{n}f, \operatorname{Im}\partial_{\bar z}^{n}f\right), \] and assume $X_f(0)=0$ and that this zero is isolated. Choose a sufficiently small $r>0$ so that $X_f$ has no zeros on $|z|=r$. Its index at the origin is the topological degree of the map $z\mapsto X_f(z)/\|X_f(z)\|$ from that positively oriented circle to the unit circle. The index is independent of the sufficiently small radius.

The question asks to prove \[ \operatorname{ind}_0(X_f)\le n. \] It concerns every analytic germ satisfying the isolated-zero condition, without assuming nondegeneracy of its Jacobian. For $n=2$ the field is $(f_{xx}-f_{yy},2f_{xy})$, illustrating the restriction imposed by being derived from one scalar function. An arbitrary analytic vector field need not satisfy the claimed bound. The source records established low-order cases; the target is the full analytic statement for general $n$. Replacing real analyticity by finite differentiability is a related extension, but is not an added requirement of this definition.

Short English statement

Bound the index of the isolated zero of an $n$th conjugate-complex derivative field by $n$.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 4700025 (AMR-046-0025), 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 25: Loewner's conjecture. Original source indexed by AMR-046-0025. https://arxiv.org/abs/2012.02524.