# Clarification regarding the definition of absolute height of an algebraic number

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  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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