Several Complex Variables Problem List — Problem 45

Definitions and mathematical statement

Let $n\geq 1$ be an integer. A domain $D\subseteq{\mathbb{C}}^n$ is a nonempty connected open set. The supplied source asks: for domains $D\subseteq{\mathbb{C}}^n$ of finite type, determine the asymptotic behavior of $k$.

This is an incompletely specified problem: the supplied record does not define $k$, its arguments, the limiting regime, or the meaning of finite type. It also does not specify boundary regularity, boundedness, or pseudoconvexity assumptions. Consequently, no precise asymptotic assertion can be formulated from this record alone; these unresolved qualifications must be recovered from the source context before mathematical review.

Short English statement

Determine the asymptotic behavior of the source's quantity $k$ on finite type domains in ${\mathbb{C}}^n$. The quantity, limiting regime, finite type convention, and necessary domain hypotheses are unspecified in the supplied record.

Sources

• Local UnsolvedMath registry: permanent ID $800246$, source problem code catalogue entry, title "Several Complex Variables Problem List --- Problem 45." The registry attributes the problem to Buzzard, Fornaess, and Krantz, with proposed year 1995, and labels it open; that status is supplied metadata, not an independently verified assessment.
• Buzzard, Fornaess, and Krantz, The Several Complex Variables Problem List (1995), Problem 45. Supplied source link: https://arxiv.org/abs/math/9509202. The record reports access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance details, not a claim of independent source consultation.