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OP-00012 Open

Schanuel's conjecture

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Given any set of $n$ complex numbers $\{z_1, ..., z_n\}$ that are linearly independent over $\mathbb{Q}$, the field extension $\mathbb{Q}(z_1, ..., z_n, e^{z_1}, ..., e^{z_n})$ has transcendence degree at least $n$ over $\mathbb{Q}$.

Source: S. Schanuel (1960s); first published in S. Lang, Introduction to Transcendental Numbers (1966), Ch. III.

Classification

Primary
NT — Number Theory
Keywords
exponential function, transcendental numbers

References

@book{lang1966,
  author = {Lang, Serge},
  title = {Introduction to Transcendental Numbers},
  publisher = {Addison-Wesley},
  year = {1966}
}

@article{ax1971,
  author = {Ax, James},
  title = {On {S}chanuel's conjectures},
  journal = {Annals of Mathematics},
  volume = {93},
  pages = {252--268},
  year = {1971}
}

@misc{wikipedia,
  title = {Schanuel's conjecture},
  howpublished = {Wikipedia},
  url = {https://en.wikipedia.org/wiki/Schanuel%27s_conjecture}
}

@misc{formalconjectures,
  author = {{The Formal Conjectures Authors}},
  title = {Formal Conjectures},
  url = {https://github.com/google-deepmind/formal-conjectures/blob/0771383387505c96d1b2f6a3d35088ad00892c5c/FormalConjectures/Wikipedia/Schanuel.lean#L33},
  note = {Lean statement under Apache-2.0; problem text under CC BY 4.0, or CC BY-SA 4.0 where it is based on Wikipedia}
}

Canonical Lean statement

A submission resolves this problem by proving the constant Problem_OP_00012. 

/-- OP-00012: Schanuel's conjecture -/
def Problem_OP_00012 : Prop :=
  ∀ (n : ℕ) (z : Fin n → ℂ),
      LinearIndependent ℚ z →
        ↑n ≤ Algebra.trdeg ℚ ↥(IntermediateField.adjoin ℚ (Set.range z ∪ Set.range (Complex.exp ∘ z)))

Formalised by The Formal Conjectures Authors (Apache-2.0); reviewed by clemens-admin.

Available to match against in: Lean v4.33.1 with Mathlib v4.33.1.

Resolving this problem in Lean

Add the problems package as a dependency, import this problem's module, and prove the constant. Any published revision works.

In lakefile.toml:

[[require]]
name = "RegistryProblems"
git = "https://github.com/cthalhammer/registry-problems"
rev = "<published revision>"

In your proof:

import RegistryProblems.OP_00012

theorem resolution : RegistryProblems.Problem_OP_00012 := by
  sorry

In preprint.toml, mapping the claim to the declaration that establishes it: 

[[statements]]
label = "Theorem 1"
lean = "resolution"
latex = "..."
problem = "OP-00012"
claim = "full"

To disprove it instead, prove the negation and claim "disproof", or "counterexample" if the proof exhibits one: 

theorem disproof : ¬ RegistryProblems.Problem_OP_00012 := by
  sorry

You can also write the statement out in full; the verifier checks it matches up to definitional equality. Partial results and reductions are not matched against this statement.

Citing this problem

This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn. 

OP-00012, "Schanuel's conjecture", clopen.solutions.