[#P15] Schanuel's conjecture

Problem. If \(z_1,\ldots,z_n\in\mathbb{C}\) are linearly independent over \(\mathbb{Q}\), then \(\operatorname{trdeg}_{\mathbb{Q}}\mathbb{Q}(z_1,\ldots,z_n,e^{z_1},\ldots,e^{z_n})\ge n\).

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Definitions and notation

1Context

This single conjecture would imply many major algebraic-independence statements, including the algebraic independence of e and pi.

2Definitions

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.

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

Saved packet · July 31, 2026

What counts as a solution

Saved packet 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]

1Packet records

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Record	Kind	Assessment
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.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 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] Relevance to this problem For Schanuel's conjecture, this successor gives readable dated status prose and the exact remaining research boundary. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 2, Reference 1 Details The packet's cited sources and equivalent formulations were checked in the dated review recorded below. 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. Open full record →
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.	Trace	Outcome: Open Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 1 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.^[1] Relevance to this problem For Schanuel's conjecture, record schanuel-conjecture-attempt-resolution-route (“Resolve the stated acceptance condition”) documents a concrete method, search boundary, or failed route. The record states: 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. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded Sources Reference 1 What happened Target the displayed statement directly. 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. Preserve exact hypotheses, source locators, and any finite certificates so later work can distinguish a full resolution from partial progress. Open full record →
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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

Merging of orbits under adding the product of nonzero digitsnumber theory
Sharp multipliers for balanced binary productsnumber theory
Prime exceptions to connectivity of the Markoff graphnumber theory

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Plain text

“Schanuel's conjecture.” TheoremDB. P15. Problem statement; statement text SHA-256 994b4246011a5de7366dec24bd0855bccd13fa6a3c6eb38301b3dcbb2d471480. https://theoremdb.org/statement/?ref=P15

BibTeX

@misc{theoremdb-problem-994b4246011a5de7366dec24bd0855bccd13fa6a3c6eb38301b3dcbb2d471480,
  title = {{Schanuel's conjecture}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 994b4246011a5de7366dec24bd0855bccd13fa6a3c6eb38301b3dcbb2d471480},
  url = {https://theoremdb.org/statement/?ref=P15}
}

Plain text: Built Markdown snapshot

This problem includes 2 records joined by 2 typed links, sourced from ams.org^[1], current as of July 31, 2026.

1Lean verification

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.

[#P15] Schanuel's conjecture

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