Abstract
In Ax (Ann. Math. 93(2):252–268, 1971), J. Ax proved a transcendency theorem for certain differential fields of characteristic zero : the differential counterpart of the still open Schanuel conjecture about the exponential function over \({\mathbb{C}}\) (Lang, Introduction to transcendental numbers, 1966). In this article, we derive from Ax’s theorem transcendency results in the context of differential valued exponential fields. In particular, we obtain results for exponential Hardy fields, Logarithmic-Exponential power series fields, and Exponential-Logarithmic power series fields.
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This joint work was inspired during the first author’s visit to Ben Gurion University sponsored by the Institute for Advanced Studies in mathematics at Ben Gurion University. The first author wishes to thank the institute for this opportunity.
The third author was supported by a postdoctoral fellowship funded by the Skirball Foundation via the Center for Advanced Studies in Mathematics at Ben-Gurion University of the Negev.
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Kuhlmann, S., Matusinski, M. & Shkop, A.C. A note on Schanuel’s conjectures for exponential logarithmic power series fields. Arch. Math. 100, 431–436 (2013). https://doi.org/10.1007/s00013-013-0520-5
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DOI: https://doi.org/10.1007/s00013-013-0520-5