Canonical Pseudo-Exponential Fields and Stability

Definitions and mathematical statement

An exponential field is a field $K$ equipped with a homomorphism $\exp:(K,+)\to(K^\times,\cdot)$. This basic definition alone does not define the pseudo-exponential fields or genericity conditions intended in the source. Let $\mathcal G$ denote the intended class of generic pseudo-exponential fields with all its source axioms; these axioms are not reproduced in the record.

The first question asks, for each uncountable cardinal $\lambda$, whether there is a canonical object $\widetilde K_{\exp}(\lambda)\in\mathcal G$ of cardinality $\lambda$. If canonical means uniqueness up to exponential-field isomorphism among the intended generic objects, the requested property is \[ \forall\lambda>\aleph_0\quad \exists K\in\mathcal G\ (|K|=\lambda) \quad\text{and}\quad \forall K,L\in\mathcal G\ (|K|=|L|=\lambda\Rightarrow K\cong L). \] This is an interpretation of canonical, not an assertion that every exponential field of that size is isomorphic or that a functorial choice has been constructed. The precise class and meaning of canonical must be recovered from the source.

The second question asks for the stability status of $\widetilde K_{\exp}$. In a first-order interpretation one would first fix the language and complete theory $T=\operatorname{Th}(\widetilde K_{\exp})$; $T$ is $\kappa$-stable if, for every parameter set $A$ of size at most $\kappa$ in a sufficiently saturated model, the space $S_1^T(A)$ of complete one-variable types over $A$ has cardinality at most $\kappa$. The record does not specify whether stability refers to that first-order theory, an infinitary axiomatization, a reduct, or a different category of structures. Uniqueness in a generic class is not by itself a first-order stability assertion.

The supplied literature triage checked 2026-08-21 says that Zilber reports answers to the canonical-choice and stability questions, but does not reproduce those answers or their hypotheses. It separately describes identification with the complex exponential field as a conjectural consequence. The natural cardinality-compatible identification would concern $\widetilde K_{\exp}(2^{\aleph_0})$ and $({\mathbb{C}},+,\cdot,\exp)$, not every uncountable size; no theorem or unmentioned hypothesis for this identification is asserted here. That identification is additional context beyond the two questions explicitly extracted.

Short English statement

Ask for a canonical generic pseudo-exponential field in each uncountable cardinality and for its stability classification. The supplied review reports answers to these two questions but leaves the precise axioms and answers unstated; identification with the complex exponential field remains only conjectural in that review.

Sources

• Local UnsolvedMath registry, permanent ID 30000099, catalogue entry, Canonical Pseudo-Exponential Fields and Stability.
• Model Theory and Complex Analytic Geometry (2005), Oberwolfach Reports, pp. 1845--1882, DOI https://doi.org/10.4171/owr/2004/35. The supplied verification identifies the first two questions in its pseudo-analytic-structures contribution.
• Zilber, supplied reference label "exponential sums": https://doi.org/10.1112/S0024610701002861.
• Supplied literature triage dated 2026-08-21, distinguishing reported answers to the two questions from a conjectural complex-exponential identification.