According to page 1-2 of this paper (https://arxiv.org/abs/0906.4286), Mahler has established the inequality $$|\alpha_1 - \alpha_2| \geq H(\alpha_1)^{-(d-1)} \tag 1 $$ to be valid for all pairs of conjugate algebraic numbers $\alpha_1$ and $\alpha_2$ of degree $d$. Here $H(\alpha)$ is, as stated therein, the "absolute height of the minimal polynomial of $\alpha$ over $\mathbb Z$".
On the other hand, the equivalent definition of absolute and logarithmic height that I am familiar with is the following:
For an algebraic integer $\alpha$ and a finite extension $K$ of $\mathbb Q$ containing $\alpha$, we define the absolute logarithmic height $h(\alpha)$ of $\alpha$ by $$h(\alpha) := \sum_{v \in M_K} \frac{[K_v:\mathbb Q_v]}{[K:\mathbb Q]} \log \max \{1, |\alpha|_v \}$$ where the sum runs over a set $M_K$ of places of $K$ satisfying the product formula and $K_v$ (respectively $\mathbb Q_v$) denotes the completion of $K$ (respectively $\mathbb Q$) with respect to the place $v \in M_K$.
This is equal to the logarithm of the Mahler measure (absolute value of the product of the conjugates lying outside he unit circle) of $\alpha$ when $\alpha$ is an algebraic integer. As far as I know, for algebraic integers $\alpha$ we can define its multiplicative height $H(\alpha)$ by $e^{h(\alpha)}$.
My question is the following: is this multiplicative height the same one as referred to by the "$H(\alpha)$" occurring in equation $(1)$ (that is, the "$H(\alpha)$" appearing in the paper linked above)? Or is that some normalized version of the multiplicative height?
The reason I ask this is that Mahler's paper (reference [18] of the attached paper) establishes that $$\delta(\alpha) > \sqrt 3 d^{-(d+2)/2} |D(\alpha)|^{1/2} M(\alpha)^{-(d-1)} \tag 2$$ where $\delta(\alpha)$ is the least distance between two conjugates of $\alpha$, $d:=\deg \alpha$ and $D(\alpha)$ is the discriminant of (the minimal polynomial of) $\alpha$; and I think what follows from this is that $|\alpha_1 - \alpha_2| > H(\alpha)^{-d^2}$, with $H(\alpha):=e^{h(\alpha)}$ being the multiplicative height defined in the previous paragraph. So either there is some error/bad estimate in my computation (in which case I would really like to know how $(1)$ exactly follows from $(2)$ with $H(\alpha)$ denoting the multiplicative height in both inequalities) or it may be possible that the $H(\alpha)$ in $(1)$ is some 'exponentiated' version of the multiplicative height (or $(1)$ could be a typo in the paper)? I would really appreciate some help or clarification. Thank you.
