Conjecture 3.8 — Gel’fond

Definitions and mathematical statement

Let $\alpha\in\overline{\mathbb Q}^{\times}$ be nonzero algebraic and choose a nonzero complex logarithm $\lambda$ of $\alpha$, so that $e^\lambda=\alpha$ and $\lambda\ne0$. Let $\beta\in\overline{\mathbb Q}\setminus\mathbb Q$ be algebraic irrational. Define the chosen value of the algebraic power by \[ \alpha^\beta:=e^{\beta\lambda}. \] The fixed logarithm is part of the data; a different logarithm can produce a different power.

Are $\lambda$ and $e^{\beta\lambda}$ algebraically independent over $\mathbb Q$ for every such choice? In explicit polynomial language, the conjecture states \[ F(\lambda,e^{\beta\lambda})\ne0 \quad\text{for every nonzero }F\in\mathbb Q[X,Y]. \] Equivalently, the field generated by these two complex numbers has transcendence degree two over $\mathbb Q$.

The algebraic-irrational hypothesis on $\beta$ and nonzero-logarithm hypothesis on $\lambda$ come from the source's surrounding setup, which the short catalogue statement omits. They are essential to the intended formulation. One does not impose an additional condition $\alpha\ne1$: the source permits $\alpha=1$ with a nonzero logarithm such as $2\pi i$.

The conclusion is a joint assertion about the logarithm and the selected power. Knowing that each entry separately is transcendental would not exclude a polynomial relation linking them. The problem quantifies over all algebraic inputs and all allowed branches, rather than a single principal-branch convention.

Short English statement

Are a nonzero logarithm of an algebraic number and its corresponding irrational algebraic power always algebraically independent?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 8700026 (AMR-086-0026), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Waldschmidt - Open Diophantine Problems (2004), Conjecture 3.8. Original source indexed by AMR-086-0026. https://arxiv.org/abs/math/0312440.
• S3 M. Waldschmidt, Open Diophantine Problems, Moscow Mathematical Journal 4 (2004), 245–305. Author PDF supplies the surrounding definitions and hypotheses. https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/odp.pdf.