Exponential algebraicity in exponential fields

Abstract

I give an algebraic proof that the exponential algebraic closure operator in an exponential field is always a pregeometry, and show that its dimension function satisfies a weak Schanuel property. A corollary is that there are at most countably many essential counterexamples to Schanuel's conjecture.

12 pages; v2 minor change to proof of lemma 6.2

Cited by in corpus (20)

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• A Schanuel property for exponentially transcendental powers
• Finitely Presented Exponential Fields
• A note on the axioms for Zilber's pseudo-exponential fields
• Generic derivations on o-minimal structures
• A geometric approach to some systems of exponential equations
• Local interdefinability of Weierstrass elliptic functions
• Excellence and uncountable categoricity of Zilber's exponential fields
• The algebraic numbers definable in various exponential fields
• A Schanuel Property for $j$
• A closure operator respecting the modular $j$-function
• Ax-Schanuel for Linear Differential Equations
• Quasiminimality of complex powers
• Independence relations for exponential fields
• Embedding the prime model of real exponentiation into o-minimal exponential fields
• Hilbert polynomials for finitary matroids
• Algebraic types in Zilber's exponential field
• Exponential ideals and a Nullstellensatz
• Differential exponential topological fields