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Algebraic types in Zilber's exponential field
Vahagn Aslanyan, Jonathan Kirby
We characterise the model-theoretic algebraic closure in Zilber's exponential field. A key step involves showing that certain algebraic varieties have finite intersections with cer…
The Existential Closedness and Zilber-Pink Conjectures
Vahagn Aslanyan
In this paper we survey the history of, and recent developments on, two major conjectures originating in Zilber's model-theoretic work on complex exponentiation -- Existential Clos…
Independence relations for exponential fields
Vahagn Aslanyan, Robert Henderson, Mark Kamsma +1
We give four different independence relations on any exponential field. Each is a canonical independence relation on a suitable Abstract Elementary Class of exponential fields, sho…
A closure operator respecting the modular -function
Vahagn Aslanyan, Sebastian Eterović, Jonathan Kirby
We prove some unconditional cases of the Existential Closedness problem for the modular -function. For this, we show that for any finitely generated field we can find a "conveni…
Differential Existential Closedness for the -function
Vahagn Aslanyan, Sebastian Eterović, Jonathan Kirby
We prove the Existential Closedness conjecture for the differential equation of the -function and its derivatives. It states that in a differentially closed field certain equati…
Existentially closed De Morgan algebras
Vahagn Aslanyan
We show that the theory of De Morgan algebras has a model completion and axiomatise it. Then we prove that it is -categorical and describe definable and algebraic closure…