Analytic germs with prescribed convex barriers

Definitions and mathematical statement

Fix $n\geq1$ and a set $A\subset\mathbb C^n$ such that every $n$ distinct vectors are linearly independent. Associate to each $a=(a_1,\ldots,a_n)\in A$ a nonempty compact convex set $K_a\subset\mathbb C$. An analytic germ at infinity vanishing there is represented by a function holomorphic for $|z|>R$ and with limit zero as $z\to\infty$; it has a convergent expansion $\sum_{m\geq1}b_mz^{-m}$ near infinity.

Characterize the assignments $a\mapsto K_a$ for which there exist $n$ such germs $F_1,\ldots,F_n$ so that each $G_a=\sum_ja_jF_j$ has the following exact continuation property. The germ extends to a single-valued holomorphic function on $\mathbb C\setminus K_a$, but for no proper compact convex subset $K\subsetneq K_a$ does it extend to all of $\mathbb C\setminus K$.

Thus $K_a$ is a minimal convex obstruction to continuing the corresponding combination. A larger excluded set would give only a weaker continuation statement and would not realize the prescribed data exactly. Degenerate convex sets such as segments or single points are permitted.

The same germs must realize all the barriers simultaneously, so cancellation among their linear combinations is central to the compatibility question. The target is a necessary and sufficient geometric and linear-algebraic criterion for arbitrary $A$ and prescribed compact sets, together with realizing germs when that criterion is satisfied.

Short English statement

Which convex singularity barriers can simultaneously belong to prescribed linear combinations of analytic germs at infinity?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3700013 (AMR-036-0013), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Eremenko - My favorite unsolved problems, Indicators, Problem 2. Original source indexed by AMR-036-0013. https://www.math.purdue.edu/~eremenko/uns1.html.
• S3 A. Eremenko, Indicators of linear combinations, Problem 2. Author-maintained primary problem note. https://www.math.purdue.edu/~eremenko/dvi/indi2.pdf.