Let $\alpha \in \overline{\mathbb{Q}}$ of degree $\leq d$ and such that $h(\alpha)\leq h$. Denote $K=\mathbb{Q}(\alpha)$.
(Here I use the definition $h(\alpha)=\frac{1}{d}\sum_{p\in spec(\mathbb{Z})}{\sum_{\sigma:K\to \mathbb{C}_{p}}}log^{+}|\sigma(\alpha)|_{p}$, where p-adic norm is taken for non-zero p and regular norm for $p=(0)$, $log^{+}x=max\{0,logx\}$, and $\mathbb{C}_{p}$ is the completion of the algebraic closure of $\mathbb{Q}_{p}$).
I wish to find a bound from above for $|\alpha|$ in terms of $d$ and $h$. To this end, I am trying to bound the coefficients of the minimal polynomial of $\alpha$, say, $f_{\alpha}\in \mathbb{Z}[x]$ is a minimal primitive polynomial for $\alpha$. The problem is that I don't know how to bound the coefficients of this $f_{\alpha}$, because I don't know how to relate the norm of $\alpha$ to the norm of its Galois conjugates.
More specifically, if $a$ is the leading coefficient of $f_{\alpha}$, then we have a formula:
$|a|\prod_{|\alpha_{i}|\geq1}|\alpha_{i}|=e^{dh}$
Where $\alpha_{i}$ are the galois conjugates of $\alpha$, which follows from the more general Jensen's formula for mahler measure.
My question: Can this formula be used to bound $|\alpha|$? If not, how would find a bound? Surely there is a reference for this question?
Thank you very much!
Best Answer
For $\beta\in \overline{\Bbb{Q}}$ let $V(\beta)$ be the set of isomorphism classes of absolute values on $\Bbb{Q}(\beta)$ with the normalization $|p|_v= p^{-1}$ if $|.|_v$ is above $\Bbb{Q}_p$
and $$h(\beta) = \frac1d\sum_{v\in V(\beta)} \max(\log|\beta|_v,0), \qquad d=[\Bbb{Q}(\beta):\Bbb{Q}]$$
The monic minimal polynomial of $\beta$ is $$f(X)=\prod_{\sigma\in Hom(\Bbb{Q}(\beta),\Bbb{C})} (X-\beta)\in \Bbb{Q}[X]$$ Since $|\beta|_{\Bbb{C}}\le e^{d\, h(\beta)}$ its coefficients are bounded by those of $$(X-e^{d\, h(\beta)})^d$$
Thus all we need is an integer $n$ such that $n f(X)\in \Bbb{Z}[X]$. We can take $$n= \prod_p\prod_{v\ |\ p, \ |\beta|_v > 1} |\beta|_v^d\le e^{d^2\ h(\beta)}$$ With this $n$ then $|n\beta|_v\le 1$ for all finite place thus $n\beta\in \overline{\Bbb{Z}}$ and $\prod_{\sigma\in Hom(\Bbb{Q}(\beta),\Bbb{C})} (X-n\beta)\in \Bbb{Z}[X]$
And hence the coefficients of the $\Bbb{Z}[X]$ minimal polynomial of $\beta$ are bounded by those of $$e^{d^2\ h(\beta)}(X-e^{d\, h(\beta)})^d$$