Finite-Dimensional L^p-Bergman Spaces on Complete Reinhardt Domains

Coordinatewise completeness imposes an arithmetic restriction on the dimension of unweighted \(L^p\)-Bergman spaces. For every positive rational \(p\), such a space on a complete Reinhardt domain in any complex dimension is either zero or infinite-dimensional. In \(\mathbb C^2\), the only possible finite positive dimension is one, and it occurs for every irrational \(p>0\). An explicit unbounded domain with smooth boundary has a one-dimensional space for a countable dense set of irrational exponents and a zero space at every other positive exponent. Its ordinary Bergman space is zero, although its logarithmic image contains no affine line. Every bounded holomorphic function on the domain is constant, and nonconstant entire curves lie on coordinate axes. For each irrational \(p\) and each integer \(k\ge2\), a related complete Reinhardt domain in \(\mathbb C^3\) has an exactly \(k\)-dimensional \(L^p\)-Bergman space.

The proofs use contained polydiscs, Taylor coefficients and explicit radial integrals. No pseudoconvexity is assumed. This is a complete scoped contribution related to AIM-ANALYSIS-0041, not a complete classification resolving its broad geometry question. Classical monomial methods and the related multiplier argument of Englis are credited. AI-assisted, self-audited and unrefereed; no independent review, formal verification or absolute priority is claimed.