Blurred Complex Exponentiation

Abstract

It is shown that the complex field equipped with the "approximate exponential map", defined up to ambiguity from a small group, is quasiminimal: every automorphism-invariant subset of the field is countable or co-countable. If the ambiguity is taken to be from a subfield analogous to a field of constants then the resulting "blurred exponential field" is isomorphic to the result of an equivalent blurring of Zilber's exponential field, and to a suitable reduct of a differentially closed field. These results are progress towards Zilber's conjecture that the complex exponential field itself is quasiminimal. A key ingredient in the proofs is to prove the analogue of the exponential-algebraic closedness property using the density of the group governing the ambiguity with respect to the complex topology.

14 pages, v2, minor changes

References in corpus (2)

• The theory of the exponential differential equations of semiabelian varieties
• Pseudo-exponential maps, variants, and quasiminimality

Cited by in corpus (5)

• Blurrings of the $j$-function
• Exponential Sums Equations and Tropical Geometry
• Differential Existential Closedness for the $j$-function
• On Some Systems of Equations in Abelian Varieties
• Equations involving the modular $j$-function and its derivatives