Conjecture 4.4 — Amoroso-David

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $\alpha=(\alpha_1,\ldots,\alpha_n)\in(\overline{\mathbb Q}^{\times})^n$. Multiplicative independence means $\prod_i\alpha_i^{a_i}=1$ with $a_i\in\mathbb Z$ implies all $a_i=0$. Define $\omega(\alpha)=\min\{\deg P:0\ne P\in\mathbb Q[X_1,\ldots,X_n],\ P(\alpha)=0\}$. For any number field $K$ containing the coordinates, the absolute logarithmic projective height is $h(1:\alpha_1:\cdots:\alpha_n)=[K:\mathbb Q]^{-1}\sum_v[K_v:\mathbb Q_v]\log\max\{1,|\alpha_1|_v,\ldots,|\alpha_n|_v\}$, with the usual normalized absolute values.

Statement recorded in the supplied source.

For every $n\ge1$, does there exist $c(n)>0$ such that every multiplicatively independent algebraic tuple $\alpha$ satisfies \[h(1:\alpha_1:\cdots:\alpha_n)\ge c(n)/\omega(\alpha)?\] [S2]

Short English statement

Is the projective height of a multiplicatively independent algebraic tuple uniformly bounded below by a constant divided by the least degree of a rational polynomial vanishing at that tuple?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 8700049 (AMR-086-0049). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Michel Waldschmidt, Open Diophantine Problems, arXiv:math/0312440v2 (24 January 2004), Conjecture 4.4 and the definitions at the start of Section 4. https://arxiv.org/abs/math/0312440.