Conjecture 3.12 — $p$-adic analog of an algebraic independence result of Gel’fond

Definitions and mathematical statement

Fix a prime $p$, work in ${\mathbb{C}}_p$ with $|p|_p=p^{-1}$, and use a fixed embedding of algebraic numbers. The power-series conventions are \[ \log_p(1+u)=\sum_{k\ge1}\frac{(-1)^{k+1}u^k}{k}\quad(|u|_p<1), \qquad \exp_p(z)=\sum_{k\ge0}\frac{z^k}{k!} \quad(|z|_p<p^{-1/(p-1)}). \] Let $\alpha\ne0$ be algebraic in the logarithm domain and let $\beta$ be cubic algebraic in the intended p-adic field. A conventional degree reading is $[{\mathbb{Q}}(\beta):{\mathbb{Q}}]=3$; whether $\beta$ must lie in ${\mathbb{Q}}_p$ rather than a larger p-adic field is not explicit.

Put $L=\log_p\alpha$. When both exponential arguments are in their domains, define \[ E_1=\alpha^\beta=\exp_p(\beta L),\qquad E_2=\alpha^{\beta^2}=\exp_p(\beta^2L). \] The intended algebraic-independence assertion is \[ \forall P\in{\mathbb{Q}}[U,V]\setminus\{0\},\qquad P(E_1,E_2)\ne0. \] The source assumes only convergence for $\beta L$. This does not by itself guarantee convergence for $\beta^2L$ when $|\beta|_p>1$; both arguments need a justified domain condition. A hypothesis such as $|\beta|_p\le1$ would suffice for that step, but is not stated and is not silently inserted.

There is also an explicit degeneracy: $\alpha=1$ satisfies the written hypotheses and gives $E_1=E_2=1$, contradicting algebraic independence. More generally $L=0$ gives this obstruction. A meaningful intended version therefore requires $\log_p\alpha\ne0$, as well as definedness of both exponentials, or an equivalent source qualification. These are flagged corrections, not replacements of the supplied literal statement. The research summary calls the intended p-adic cubic independence problem open and compares it with a classical complex Gel'fond result, without resolving these missing hypotheses.

Short English statement

The intended conjecture asserts algebraic independence of two p-adic exponentials with cubic exponent parameter. As written it allows $\alpha=1$ and does not guarantee that the second exponential converges, so its nondegeneracy and domain hypotheses need repair.

Sources

• Local UnsolvedMath registry, permanent ID $8700030$, exact source code catalogue entry, "Conjecture 3.12 --- p-adic analog of an algebraic independence result of Gel'fond." Proposer Michel Waldschmidt.
• Waldschmidt, Open Diophantine Problems (2004), Conjecture 3.12; https://arxiv.org/abs/math/0312440. Source-TeX extraction; recorded access 2026-07-29.
• Supplied OPEN-TRIAGE research summary describes the intended p-adic question as unresolved and mentions the classical complex Gel'fond theorem, but gives no further precise citation or missing-domain clarification.