Conjecture 3.10 — Gel’fond-Schneider

Definitions and mathematical statement

Let $\alpha\in\overline{\mathbb Q}^{\times}$, and choose a nonzero complex logarithm $\lambda$ with $e^\lambda=\alpha$. Let $\beta$ be algebraic irrational of degree $d=[\mathbb Q(\beta):\mathbb Q]\ge2$. With this one logarithm fixed, define $\alpha^z=e^{z\lambda}$ for every complex $z$.

The combined Gelfond--Schneider conjecture asks whether the following $d$ complex numbers are algebraically independent over $\mathbb Q$: \[ \lambda,\quad e^{\beta\lambda},\quad e^{\beta^2\lambda}, \quad\ldots,\quad e^{\beta^{d-1}\lambda}. \] Explicitly, evaluating any nonzero polynomial in $d$ variables with rational coefficients at this ordered list should give a nonzero complex number. Equivalently, the field they generate over $\mathbb Q$ should have transcendence degree $d$.

The quantification is over every permitted algebraic pair and every nonzero logarithm of $\alpha$. No principal-branch restriction is imposed. The possibility $\alpha=1$ is included when the chosen logarithm is nonzero. These input conditions belong to the original source's setup and are needed to make the catalogue's abbreviated statement complete.

The power in the $j$th exponential is $\beta^j\lambda$, so all entries must use the same $\lambda$. The conclusion simultaneously excludes relations among the exponential entries and relations also involving the logarithm. Proving algebraic independence after omitting $\lambda$, or proving independence of each pair containing $\lambda$, would not by itself establish the full assertion for general $d$.

Short English statement

Are a chosen nonzero algebraic logarithm and all the indicated powers formed from it jointly algebraically independent?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 8700028 (AMR-086-0028), 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 3.10. Original source indexed by AMR-086-0028. 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.