Exponentially Closed Fields and the Conjecture on Intersections with Tori

Abstract

We give an axiomatization of the class ECF of exponentially closed fields, which includes the pseudo-exponential fields previously introduced by the second author, and show that it is superstable over its interpretation of arithmetic. Furthermore, ECF is exactly the elementary class of the pseudo-exponential fields if and only if the diophantine conjecture CIT on atypical intersections of tori with subvarieties is true.

27 pages, substantial improvements to the presentation

References in corpus (9)

• On Quasiminimal Excellent Classes
• Exponential algebraicity in exponential fields
• The theory of the exponential differential equations of semiabelian varieties
• Quasiminimal structures and excellence
• Covers of Multiplicative Groups of Algebraically Closed Fields of Arbitrary Characteristic
• Finitely Presented Exponential Fields
• A note on the axioms for Zilber's pseudo-exponential fields
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Cited by in corpus (13)

• Pseudo-exponential maps, variants, and quasiminimality
• A note on the axioms for Zilber's pseudo-exponential fields
• A geometric approach to some systems of exponential equations
• Weak Modular Zilber-Pink with Derivatives
• Blurred Complex Exponentiation
• Tropical Varieties for Exponential Sums
• Generic Solutions of Equations Involving the Modular $j$-function
• Independence relations for exponential fields
• Algebraic Varieties and Automorphic Functions
• Solutions of equations involving the modular $j$ function
• Classification over a predicate -- the general case. Part I -- structure theory
• Algebraic types in Zilber's exponential field
• Model theory of special subvarieties and Schanuel-type conjectures