Zilber constructed a unique 'pseudo-exponential' field B\mathbb{B} of continuum cardinality satisfying algebraic analogues of the key properties of the complex exponential field Cexp⁡=(C,+,⋅,exp⁡)\mathbb{C}_{\exp}=(\mathbb{C},+,\cdot,\exp), and conjectured that Cexp⁡≅B\mathbb{C}_{\exp}\cong\mathbb{B}; this would imply Schanuel's conjecture from transcendence theory. Kirby proved that the conjecture is equivalent to the a priori weaker statement that Cexp⁡\mathbb{C}_{\exp} and B\mathbb{B} are elementarily equivalent (satisfy the same first-order sentences), and identified the unconditional obstacles: it is unknown whether exponential-algebraic closedness coincides with strong exponential-algebraic closedness without assuming the diophantine conjecture CIT (on intersections with tori), and neither an unconditional description of the saturated models of the common first-order theory TBT_{\mathbb{B}} nor an unconditional quantifier-elimination result for TBT_{\mathbb{B}} is known. These questions are all open as of 2026.

Background

An exponential field is a field FF of characteristic zero equipped with a group homomorphism exp⁡:F→F×\exp:F\rightarrow F^{\times} from the additive to the multiplicative group; its kernel is ker⁡(exp⁡)={x:exp⁡(x)=1}\ker(\exp)=\{x:\exp(x)=1\}. The central example is the complex exponential field Cexp⁡=(C,+,⋅,exp⁡)\mathbb{C}_{\exp}=(\mathbb{C},+,\cdot,\exp), whose kernel is 2πiZ2\pi i\mathbb{Z}. Schanuel's conjecture predicts that for any complex numbers z1,…,znz_1,\dots,z_n that are linearly independent over Q\mathbb{Q}, the field Q(z1,…,zn,ez1,…,ezn)\mathbb{Q}(z_1,\dots,z_n,e^{z_1},\dots,e^{z_n}) has transcendence degree at least nn over Q\mathbb{Q}; it is open, and its consequences for transcendence theory would be drastic (for example, it implies the algebraic independence of ee and π\pi).

Motivated by this and by the model-theoretic tameness of Cexp⁡\mathbb{C}_{\exp} that Schanuel's conjecture would entail, Zilber (2005) defined a class of pseudo-exponential fields: ELA-fields — exponential fields whose underlying field is algebraically closed of characteristic zero and whose exponential homomorphism exp⁡:F→F×\exp:F\rightarrow F^{\times} is surjective — with kernel τZ\tau\mathbb{Z} for a single transcendental τ\tau (the standard-kernel condition), satisfying the Schanuel property (the conclusion of Schanuel's conjecture, stated as the predimension inequality tdQ(zˉ,ezˉ)−ldimQ(zˉ)≥0\mathrm{td}_{\mathbb{Q}}(\bar z,e^{\bar z})-\mathrm{ldim}_{\mathbb{Q}}(\bar z)\geq 0 for all finite tuples zˉ\bar z, where tdQ\mathrm{td}_{\mathbb{Q}} is the transcendence degree over Q\mathbb{Q} of the field generated by the listed elements and ldimQ\mathrm{ldim}_{\mathbb{Q}} is the dimension of the Q\mathbb{Q}-vector space they span), satisfying the countable closure property (the exponential-algebraic closure of any countable set is countable), and satisfying strong exponential-algebraic closedness (SEAC), an axiom scheme asserting that every rotund, additively and multiplicatively free algebraic subvariety of dimension nn in Fn×(F×)nF^n\times(F^{\times})^n contains a generic point of the graph of exponentiation — 'rotund' and 'free' being the dimension-theoretic non-degeneracy conditions saying the variety has no trivial reason (a coordinate-projection degeneration or an additive/multiplicative subgroup relation among coordinates) to miss the graph. Exponential-algebraic closedness (EAC) is Zilber's weaker version of the same closure scheme, dropping the genericity demand on the witness point. Zilber proved (with gaps later filled by Bays and Kirby) that this quasiminimal excellent class — 'quasiminimal' meaning that every definable subset of the universe is countable or co-countable, and 'excellent' being a technical condition on the associated pregeometry (closure operator) that lets the whole class be axiomatized by an Lω1,ωL_{\omega_1,\omega}-sentence — is categorical in every uncountable cardinality; its unique model of size continuum is denoted B\mathbb{B}, and Zilber conjectured Cexp⁡≅B\mathbb{C}_{\exp}\cong\mathbb{B} — a statement which, since the Schanuel property holds in B\mathbb{B}, implies Schanuel's conjecture itself.

Kirby's note (arXiv:1006.0894; Notre Dame J. Formal Logic, 2013) analyzed the axioms and the first-order theory TBT_{\mathbb{B}} of the pseudo-exponential fields. Its Theorem 1 shows that if Cexp⁡\mathbb{C}_{\exp} is elementarily equivalent to B\mathbb{B} (written Cexp⁡≡B\mathbb{C}_{\exp}\equiv\mathbb{B}, meaning both structures satisfy exactly the same first-order sentences in the language {+,⋅,exp⁡}\{+,\cdot,\exp\}), then in fact Cexp⁡≅B\mathbb{C}_{\exp}\cong\mathbb{B}; so the isomorphism conjecture reduces to elementary equivalence, though Kirby stresses this gives no method to prove it. The note also isolates the unconditional obstacles. First, Kirby and Zilber had proved that EAC and SEAC coincide (given the remaining axioms) assuming CIT, the conjecture on intersections with tori — a diophantine geometric conjecture about atypical intersections of algebraic subvarieties of tori with subgroups (closely related to the Zilber–Pink/Mordell–Lang family) — and Kirby asks whether this coincidence can be proved without CIT. Second, without CIT there is no description of the saturated models of TBT_{\mathbb{B}} (a saturated model is one realizing many types, the standard setting for quantifier elimination; quantifier elimination means every definable set is definable by a quantifier-free formula), so no unconditional quantifier-elimination theorem for TBT_{\mathbb{B}} is available — Kirby's proofs route around this via a direct analysis of the types realized in pseudo-exponential fields. The axioms themselves are first-order except that the standard-integers condition (axiom 2c, omitting the type of a nonstandard integer) is inherently an Lω1,ωL_{\omega_1,\omega}-sentence (a countable conjunction of first-order conditions, an infinitary logic not reducible to first-order axioms by compactness), and the countable closure property needs the 'there exist uncountably many' quantifier QQ.

Later work has steadily firmed up the pseudo side and made partial progress on the complex side, but never closed the gap. Bays and Kirby (2013) completed the excellence and uncountable categoricity proof for Zilber's class. Bays and Kirby (2018) extended the pseudo-analytic construction to abelian varieties and showed that there too the complex analytic structure is isomorphic to its pseudo-analytic counterpart if and only if the corresponding Schanuel-type conjecture holds. On the complex side, Zilber's Exponential Closedness conjecture (a solvability criterion for systems of polynomial-exponential equations, far weaker than the Schanuel property) remains open in general, with growing families of special cases proved (exponential sums, raising to powers, work of Gallinaro, Mantova–Masser, and others surveyed by Aslanyan and Gallinaro in 2024), and Wilkie (2023) proposed a strategy, not yet a proof, even for the weaker quasiminimality conjecture for Cexp⁡\mathbb{C}_{\exp} (every definable subset of C\mathbb{C} is countable or co-countable). Most recently, Aslanyan and Kirby (2025) characterized model-theoretic algebraic closure in B\mathbb{B}, noting that B\mathbb{B} is not even a homogeneous model of TBT_{\mathbb{B}} and that the corresponding questions about Cexp⁡\mathbb{C}_{\exp} — including algebraic closure there — remain completely open, with Zilber's isomorphism conjecture still standing as the field's central target.

Problem Statement

Determine whether the complex exponential field Cexp⁡=(C,+,⋅,exp⁡)\mathbb{C}_{\exp}=(\mathbb{C},+,\cdot,\exp) is elementarily equivalent to Zilber's pseudo-exponential field B\mathbb{B} — that is, whether Cexp⁡\mathbb{C}_{\exp} and B\mathbb{B} satisfy exactly the same first-order sentences in the language {+,⋅,exp⁡}\{+,\cdot,\exp\}. By Kirby's Theorem 1 this is equivalent to Zilber's isomorphism conjecture Cexp⁡≅B\mathbb{C}_{\exp}\cong\mathbb{B}, so the problem may equivalently be answered by exhibiting such an isomorphism, or by refuting elementary equivalence via an explicit first-order sentence true in one structure and false in the other. Three further questions are open parts of the same objective of an unconditional first-order structural theory of pseudo-exponentiation: (1) prove or disprove that, given the remaining axioms (algebraically closed exponential field of characteristic zero with standard kernel and the Schanuel property), exponential-algebraic closedness coincides with strong exponential-algebraic closedness without assuming the diophantine conjecture CIT on intersections with tori; (2) describe unconditionally the saturated models of the common first-order theory TBT_{\mathbb{B}} of the pseudo-exponential fields; (3) give an unconditional quantifier-elimination theorem for TBT_{\mathbb{B}}. An answer to the main question (in either direction) resolves the problem; the three sub-questions are independent open problems in the same objective, are not required for that resolution, and each may be answered separately.

References

  1. Jonathan Kirby, 'A Note on the Axioms for Zilber's Pseudo-Exponential Fields', Notre Dame Journal of Formal Logic 54 (2013), DOI: 10.1215/00294527-2143844, arXiv:1006.0894, https://arxiv.org/abs/1006.0894
  2. Boris Zilber, 'Pseudo-exponentiation on algebraically closed fields of characteristic zero', Annals of Pure and Applied Logic 132 (2005), 67-95, DOI: 10.1016/j.apal.2004.07.001, https://doi.org/10.1016/j.apal.2004.07.001
  3. Martin Bays, Jonathan Kirby, 'Excellence and uncountable categoricity of Zilber's exponential fields', arXiv:1305.0493 (2013), https://arxiv.org/abs/1305.0493
  4. Jonathan Kirby, Boris Zilber, 'Exponentially Closed Fields and the Conjecture on Intersections with Tori', Annals of Pure and Applied Logic 165 (2014), 1680-1706, DOI: 10.1016/j.apal.2014.06.002, arXiv:1108.1075, https://arxiv.org/abs/1108.1075
  5. Martin Bays, Jonathan Kirby, 'Pseudo-exponential maps, variants, and quasiminimality', Algebra & Number Theory 12 (2018), 493-549, DOI: 10.2140/ant.2018.12.493, arXiv:1512.04262, https://arxiv.org/abs/1512.04262
  6. Jonathan Kirby, 'The rational field is not universally definable in pseudo-exponentiation', Fundamenta Mathematicae 232 (2016), 79-88, DOI: 10.4064/fm232-1-6, arXiv:1407.1068, https://arxiv.org/abs/1407.1068
  7. Vincenzo Mantova, 'A pseudoexponential-like structure on the algebraic numbers', Journal of Symbolic Logic 80 (2015), 1339-1347, DOI: 10.1017/jsl.2014.41, arXiv:1202.4022, https://arxiv.org/abs/1202.4022
  8. Paola D'Aquino, Antongiulio Fornasiero, Giuseppina Terzo, 'A weak version of the Strong Exponential Closure', arXiv:1909.10241 (2019), https://arxiv.org/abs/1909.10241
  9. Alex Wilkie, 'Analytic continuation and Zilber's quasiminimality conjecture', arXiv:2306.14562 (2023), https://arxiv.org/abs/2306.14562
  10. Jonathan Kirby, 'Around Zilber's quasiminimality conjecture', arXiv:2306.05811 (2023), https://arxiv.org/abs/2306.05811
  11. Vahagn Aslanyan, Jonathan Kirby, 'Algebraic types in Zilber's exponential field', Model Theory 4 (2025), 37-54, DOI: 10.2140/mt.2025.4.37, arXiv:2405.01399, https://arxiv.org/abs/2405.01399
  12. Vahagn Aslanyan, Francesco Gallinaro, 'Exponential sums equations and the Exponential Closedness conjecture', arXiv:2409.12860 (2024), https://arxiv.org/abs/2409.12860
  13. Ali Bleybel, 'On Zilber's field', arXiv:1310.3777 (2013, unpublished preprint), https://arxiv.org/abs/1310.3777