Conjecture 4.14

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.

For an algebraic number $\alpha$, choose its primitive irreducible polynomial $f_\alpha\in\mathbb Z[T]$ with positive leading coefficient $a$ and roots $\alpha^{(1)},\ldots,\alpha^{(d)}$. Set \[ M(\alpha)=a\prod_{j=1}^d\max\{1,|\alpha^{(j)}|\},\qquad \mathrm h(\alpha)=\frac{\log M(\alpha)}d, \] and let $H(\alpha)$ be the maximum absolute value of the coefficients of $f_\alpha$. These are respectively Mahler measure, absolute logarithmic height, and usual polynomial height.

Let $m\ge1$, let $\lambda_i\in\mathbb C$ satisfy $\alpha_i=e^{\lambda_i}\in\overline{\mathbb Q}^{\times}$, and let $\beta_0,\ldots,\beta_m\in\overline{\mathbb Q}$. Put $D=[\mathbb Q(\alpha_1,\ldots,\alpha_m,\beta_0,\ldots,\beta_m):\mathbb Q]$. Thus logarithms are chosen complex logarithms, with no restriction to a principal branch.

Statement recorded in the supplied source.

Do there exist absolute constants $c_1,c_2>0$ such that, for every such tuple and every real $h$ satisfying \[ h\ge \frac1D,\quad h\ge\max_i\mathrm h(\alpha_i),\quad h\ge\frac1D\max_i|\lambda_i|,\quad h\ge\max_{0\le j\le m}\mathrm h(\beta_j), \] both assertions hold?

1. If $\Lambda=\beta_0+\sum_{i=1}^m\beta_i\lambda_i\ne0$, then $|\Lambda|\ge e^{-c_1mD^2h}$.
2. If $\lambda_1,\ldots,\lambda_m$ are linearly independent over $\mathbb Q$, then $\sum_{i=1}^m|\lambda_i-\beta_i|\ge e^{-c_2mD^{1+1/m}h}$.

[S2]

Short English statement

Are these two uniform quantitative lower bounds valid for nonzero algebraic linear forms in logarithms and for simultaneous approximation of independent logarithms by algebraic numbers?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 8700056 (AMR-086-0056). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Michel Waldschmidt, Open Diophantine Problems, arXiv:math/0312440v2 (24 January 2004), Conjecture 4.14, pp. 40--41; height conventions, p. 35. https://arxiv.org/abs/math/0312440v2.