Generic solutions of exponential polynomials over \(\mathbb{C}\)

Giuseppina Terzo (Seconda Università di Napoli)


Zilber in [1] proved that the class of exponential algebraically closed
fields of characteristic 0, satisfying a natural \( \mathcal{L}_{\omega_1,\omega}(\mathbb{Q})\) sentence, expressing
strong exponential closure, countable closure and cyclicity of the kernel of
exponentiation, has a unique model in every uncountable cardinality. He
conjectured that the complex exponential field is the unique such model of
cardinality \(2^{\aleph_0}\): Marker in [2] investigated a simple case of the strong ex-
ponential closure axiom, assuming Schanuel's Conjecture. Following this
line of research, we examine the next natural cases of the strong exponen-
tial closure axiom for the complex exponential field. Assuming Schanuel's
Conjecture, we prove that certain exponential polynomials have always a
solution in \(\mathbb{C}\) which is generic. In fact, we hope to prove that (we have
only partial results) there are infinitely many algebraically independent

[1] B. Zilber: 'Pseudo-exponentiation on algebraically closed fields of charac-
teristic zero', Annals of Pure and Applied Logic, (1) 132, (2004), 67-95.
[2] D. Marker: 'A remark on Zilber's pseudoexponentation,' Journal of Sym-
bolic Logic, (3), 71, (2006), 791-798.

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