Decidability of Diophantine Equations over Rings of Holomorphic Functions via O-Minimal Geometry

We investigate the decidability of Diophantine equations over rings of holomorphic functions, bridging the gap between classical undecidability results in arithmetic geometry and the tame topology of o-minimal structures. While Hilbert's Tenth Problem is known to be undecidable for the ring of global entire functions O(C) due to the definability of the integers, we demonstrate that restricting the domain to functions definable in o-minimal expansions of the real field (such as Ran, exp) restores decidability. We construct a model-theoretic framework utilizing the Peterzil-Starchenko theory of definable complex analysis to prove that the existential theory of the ring of definable holomorphic germs is decidable. Conversely, we analyze the boundary of this "tameness" by characterizing the growth conditions under which the ring of entire functions recovers the full arithmetic complexity of Z, thereby delineating the precise logical threshold between decidable geometry and undecidable arithmetic in the context of complex analysis.