On a theorem of Forelli in a non-standard setting
Volume 128 / 2024
Banach Center Publications 128 (2024), 127-134
MSC: Primary 32A05; Secondary 32A10, 14P15, 14P20
DOI: 10.4064/bc128-8
Abstract
By a classical theorem of Forelli, a complex-valued function on the open unit ball in is holomorphic if it is holomorphic along each vector line and satisfies some weak differentiability conditions near the origin. Given any uncountable real closed field R (possibly non-Archimedean), we prove an analogous result for functions defined over the algebraic closure C = R(\sqrt {-1}) of R. We also give an example showing that uncountability of R is essential.
