[#P15] Schanuel's conjecture
Problem. If \(z_1,\ldots,z_n\in\mathbb{C}\) are linearly independent over \(\mathbb{Q}\), then \(\operatorname{trdeg}_{\mathbb{Q}}\mathbb{Q}(z_1,\ldots,z_n,e^{z_1},\ldots,e^{z_n})\ge n\).
1Context
This single conjecture would imply many major algebraic-independence statements, including the algebraic independence of e and pi.
2Problem setup
Definition 1 (Linear independence over Q). Linear independence over Q means that no nonzero rational linear combination of the z_i equals zero.
Definition 2 (The transcendence degree). The transcendence degree is the largest number of algebraically independent elements in the generated field.
Remark 1. This single conjecture would imply many major algebraic-independence statements, including the algebraic independence of e and pi.
3What counts as a solution
- Prove the stated transcendence-degree bound for every finite rationally independent tuple of complex numbers, or exhibit a tuple for which the relevant algebraic relations force smaller transcendence degree.
1Status
Current status (Dated status and exact unresolved remainder). Unresolved in this packet after the dated source check. Strongest checked result: The n=1 and algebraic-input cases follow from classical transcendence theorems, and Ax proved a functional analogue. These do not prove the full numerical conjecture. Exact unresolved remainder: Prove the transcendence-degree bound for every finite Q-linearly independent complex tuple, or exhibit a violating tuple.[2][1]
1Records
Notes and companion material
Original intake status. The cited AMS publication identifies Schanuel's conjecture as open. The source and public status were checked on 2026-07-31. This is an admin-curated seed record, not an independent exhaustive literature review.
- Functional analogues and many consequences are known. Consult current transcendence and model-theory literature before claiming a special case.
2See also
How to cite
TheoremDB contributors, “Schanuel's conjecture,” TheoremDB research memory, snapshot of July 31, 2026. https://theoremdb.org/statements/schanuel-conjectureThis page as plain text: schanuel-conjecture.md
This problem includes 2 records joined by 2 typed links, sourced from ams.org[1], current as of July 31, 2026.
1References
- Packet source. AMS Current Events Bulletin 2022, source checked for the TheoremDB status review (2026-07-31). Discussion of Schanuel's conjecture in the AMS Current Events Bulletin 2022 collection. ↗website · primary source · checked 2026-07-31Source use: original summary.The cited AMS publication identifies Schanuel's conjecture as open. The source and public status were checked on 2026-07-22. This is an admin-curated seed record, not an independent exhaustive literature review.Also cited at Discussion of Schanuel's conjecture.Also cited at Editorial research route recorded 2026-07-31.Source used to formulate or check the problem record.Source used to assess the problem's recorded status.Packet-linked open-status discussion.Source named by the research packet.
- Jonathan Kirby, “Variants of Schanuel's conjecture”. arXiv:1801.08765 (2018). Abstract and dependency survey. ↗preprint · primary source · arXiv:1801.08765v1 · checked 2026-08-01Source use: original summary.Catalogs variants and known dependencies.
An original CC0 restatement prepared by TheoremDB maintainers.