In this paper from 1980 K. Győry proved an upper bound for the absolute value of the solutions to a given Thue-Mahler equation, but I don't understand what he meant with $\log^3$.
Namely, consider a number field $L$ of degree $\ell \geq 1$. In Corollary 1, page 5, he gives bounds for the solutions of a bivariate Thue-Mahler equation $$ f(x,y) = \beta \pi_1^{z_1} \dotsb \pi_s^{z_s} \quad \text{in } x,y \in \mathcal{O}_L \text{ s.t. } N((x,y)) \leq d \text{ and } z_1,\dotsc,z_s \in \Bbb{Z}_{\geq 0} \tag{1} \label{eq:1} $$ with $d \geq 1$ a rational integer, $\beta,\pi_1,\dotsc,\pi_s$ non-zero algebraic integers in $L$ where $\pi_1,\dotsc,\pi_s$ are non-units such that $(\pi_i)$ are powers of prime ideals, and $f(X,Y) \in \mathcal{O}_L[X,Y]$ such that:
- $\deg f \geq 3$,
- $f(1,0) \neq 0$, and
- $f(X,1)$ has at least three distinct roots.
Those bounds are defined (multiplicatively) in terms of the constant $$ T^* = \exp\left(c_{f,\beta,\pi_1,\dotsc,\pi_s} (\log P)^5 \log^3(R_G^* h_G)\right) $$ where:
- $P \geq \max\operatorname{rad}(N(\pi_i)) \geq 2$,
- $c_{f,\beta,\pi_1,\dotsc,\pi_s}>0$ is a constant depending on $c_{f,\beta,\pi_1,\dotsc,\pi_s}$,
- $h_G$ is the class number of the splitting field $G$ of $f$, and
- $R_G^* = \max(R_G, e)$, with $R_G$ the regulator of $G$.
Here's my problem: Since both $(\log P)^5$ and $\log^3(R_G^* h_G)$ appear in the same formula, I assume that $$ \log^3(R_G^* h_G) \neq \log(R_G^* h_G) \log(R_G^* h_G) \log(R_G^* h_G) $$ which, as far as I know, leaves only $$ \log^3(R_G^* h_G) = \log\log\log(R_G^* h_G). $$ But consider for example $L = \Bbb{Q}$, $f = X^4 - Y^4$, $\beta = 1$, $\pi_1 = 2$, and $\pi_2 = 5$. Then $f(X,1)$ has $4$ distinct roots over its splitting field $G = \Bbb{Q}[i]$ so, as far as I understand, we are in the hypotheses of Corollary 1. However, G has regulator $R_G = 1$ and class number $h_G = 1$, so $\log\log\log(R_G^* h_G) = \log\log\log(e)$ isn't defined. Furthermore, even interpreting $$ \log\log\log(e) = -\infty $$ (over the extended reals) doesn't lead me anywhere: then $T^* = 0$ would give the upper bound $$ 2^{z_1} \cdot 5^{z_2} \leq 0 $$ for every solution $(x,y,z_1,z_2)$ of $\eqref{eq:1}$, i.e. that the equation doesn't have any solution. But this isn't the case, because $$ 3^4 - 1^4 = 80 = 2^4 \cdot 5. $$