# P15: Schanuel's conjecture

- ID: `P15`
- Reference: `schanuel-conjecture`
- Page: https://theoremdb.org/statements/P15
- Record maturity: Reviewed problem with recorded work

## Problem

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

### Context

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

### Problem setup

- **Definition (Linear independence over Q).** Linear independence over Q means that no nonzero rational linear combination of the z_i equals zero.
- **Definition (The transcendence degree).** The transcendence degree is the largest number of algebraically independent elements in the generated field.
- **Remark.** This single conjecture would imply many major algebraic-independence statements, including the algebraic independence of e and pi.

### What 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.

## Status

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](#reference-2) [1](#reference-1)

## Work

### Evidence for the current status

**Claim 1 (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.

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.

### Background and intake notes

- Original intake status: 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.
- The formulation and status were checked against the cited AMS publication on 2026-07-22.
- Functional analogues and many consequences are known. Consult current transcendence and model-theory literature before claiming a special case.

### Open directions

- **Route 1** (reported): 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](#reference-1)

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `schanuel-conjecture`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>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 https://www.ams.org/meetings/lectures/2022-CEB-master-EBOOK-3-30-22.pdf
   - Also cited at Discussion of Schanuel's conjecture
   - Also cited at Editorial research route recorded 2026-07-31
   - website; primary source; checked 2026-07-31
   - Source 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.
   - 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.
2. <a id="reference-2"></a>Jonathan Kirby, “Variants of Schanuel's conjecture”. arXiv:1801.08765 (2018). Abstract and dependency survey https://arxiv.org/abs/1801.08765
   - preprint; primary source; arXiv:1801.08765v1; checked 2026-08-01
   - Source use: original_summary
   - Catalogs variants and known dependencies.