MathOverflow Asked on January 3, 2022

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.

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