| Résume | Zilber conjectured that the complex field equipped with the exponential function is quasiminimal: every definable subset of the complex numbers is countable or co-countable. If true, it would mean that the geometry of solution sets of complex exponential-polynomial equations and their projections is somewhat like algebraic geometry. If false, it is likely that the real field is definable and there may be no reasonable geometric theory of these definable sets.
I will report on some progress towards the conjecture, including a proof when the exponential function is replaced by the approximate version given by ∃ q,r in Q [y = e^x+q+2πi r]. This set is the graph of the exponential function blurred by the group exp(Q + 2 πi Q). We can also blur by a larger subgroup and prove a stronger version of the theorem. Not only do we get quasiminimality but the resulting structure is isomorphic to the analogous blurring of Zilber's exponential field and to a reduct of a differentially closed field.
Reference: Jonathan Kirby, Blurred Complex Exponentiation, arXiv:1705.04574
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