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%% An elementary approach to effective diophantine approximation on GG_m
%% Enrico Bombieri and Paula B. Cohen
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\title{An elementary approach to effective diophantine approximation on
$\G_m$}
\author{Enrico Bombieri and Paula B. Cohen}
\date{\em To Sir Peter Swinnerton-Dyer, on his 75th birthday}
% [\hfill E. Bombieri and P. B. Cohen]
% \address{Enrico Bombieri\\
% School of Mathematics\\
% Institute for Advanced Study\\
% Princeton, N.J. 08540, USA\\ }
% \email{eb@math.ias.edu}
% \ \\
% \address{Paula B. Cohen\\
% Department of Mathematics\\
% Texas A\&M University\\
% College Station, TX 77843, USA\\ }
% \email{pcohen@math.tamu.edu}
\setcounter{page}{41}
\begin{document}
\maketitle
\addcontentsline{toc}{chapter}{Enrico Bombieri and Paula B. Cohen,
Effective diophantine approximation on $\G_m$}
% \tableofcontents
\markboth{\qquad Effective diophantine approximation on $\G_m$
\hfill}{\hfill Enrico Bombieri and Paula B. Cohen \qquad}
%%%%%%%%%%%%%%%%%%%%%%%%%% SECTION 1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
Effective results in the diophantine approximation of algebraic numbers are
difficult to obtain, and for a long time the only general method
available was Baker's theory of linear forms in logarithms. An
alternative, more algebraic, method was later proposed in Bombieri
\cite{Bo} and Bombieri and Cohen \cite{BoCo}. This new method is quite
different from the classical
approach through the theory of linear forms in logarithms.
In this paper, we improve on results derived in \cite{BoCo}. These results
concern effective approximations to roots of high order of algebraic numbers
and their application to diophantine approximation in a number field by a
finitely generated multiplicative subgroup. We restrict our attention to
the non-archimedean case, although our results and methods should
go over {\it mutatis mutandis} to the archimedean setting.
We do not claim that our theorems are the best that are known in this
direction. Linear forms in two logarithms (which are
easier to treat than the general case) suffice to prove
somewhat better results than our Theorem~\ref{1}, see Bugeaud \cite{Bu} and
Bugeaud and Laurent \cite{BuLa}; we give an explicit comparison
in \S 5, Remark \ref{Rem}.
Theorem~\ref{2}, which is useful for general applications, follows from
Theorem~\ref{1} by means of a trick introduced for the first time
in \cite{Bo} and improved in \cite{BoCo}. Thus any improved form of
Theorem~\ref{1} carries automatically an improvement of Theorem~\ref{2}.
Note however that Theorem~\ref{2}, in the form given here, is still far
from what one can obtain directly from Baker's theory of logarithmic
forms in many variables, as in Baker and W\"ustholz \cite{BW} in the
archimedean case and Kunrui Yu \cite{Yu,Yu2} in the $p$-adic case.
Notwithstanding the comparison with Baker's theory, we feel that there is
some untapped potential here. For example, one treats with equal ease
the archimedean and the $p$-adic case, while this is not so in
Baker's theory because of the bad analytic behaviour of the $p$-adic
exponential.
The auxiliary construction involves a universal family of two-variable
polynomials invariant under an action of roots of unity of a
certain order. The main new feature in the current paper is the
use of an elementary Wronskian argument, involving
differentiation only in a single variable, to derive a zero
estimate which bypasses former appeals to a more sophisticated
two-variable Dyson's Lemma. This was initially inspired by
private communication between the first author and David Masser
in 1984. We reproduce part of that communication in \S6.
Although the method of the current paper is more elementary, the
results obtained are sharper than those of \cite{BoCo}.
The main results are stated in \S5, Theorem~\ref{1} and Theorem~\ref{2}.
Theorem~\ref{1} represents an improvement over the corresponding result of
\cite{BoCo} both in the absolute constants and in the lower bound
for $r$ in (H1), where $(\log{\frac 1\kappa})^7$ is replaced by
$(\log{\frac 1\kappa})^5$, as well as in the lower bound for
$h(\alpha')$ in (H2) of \cite{BoCo}, which is no longer required. These
improvements automatically carry over to Theorem~\ref{2}, which we
restate for convenience here in the Main Theorem. We follow the
notations of \cite{BoCo}, \S2. In particular $H(\;)$ denotes the
absolute Weil height, $h(\;)=\log H(\;)$ the absolute logarithmic Weil
height and $|\phantom{a}|_v$ is the absolute value associated to a place
$v\in M_K$, normalized so that $h(x)=\sum_{v\in M_K}\max(0,\log|x|_v)$.
We define $\rho(x)$ to be the solution $\rho(x)>e^5$
of $\rho/(\log \rho)^5=x$ if $x>e^55^{-5}$, and $\rho(x)=e^5$ otherwise;
for large $x$ we have $\rho(x)\sim x(\log x)^5$.
\begin{mth} Let $K$ be a number field of degree $d$ and $v$
a place of $K$ dividing a rational prime $p$. We denote
by $f_v$ the residue class degree of the extension $K_v/\Q_p$ and
set $D_v^* = \max(1, \frac{d}{f_v\log p})$. Define a modified logarithmic
height of $x\in K$ by $h'(x) = \max\bigl(h(x),\frac{1}{D_v^*}\bigr)$, and
let $H'(x)=\exp h'(x)$.
Let $\Gamma$ be a finitely generated subgroup of the multiplicative group
$K^*$, and write $\xi_1,\dots,\xi_t$ for generators of\/
$\Gamma/\mathrm{tors}$. Let $\xi\in\Gamma$, $A\in K^*$ and $\kappa>0$
be such that
\[
0<|1-A\xi|_v A\ge h(a),\enspace G\ge h(\gamma),\enspace r=\rho A,\enspace s+1=\sigma A\le r,
\enspace n+1=\nu G, \tag{A2}
\end{equation}
and set
\[
\lambda\log\rho=\Lambda;
\]
decreasing $\lambda$ if needed, we also assume that
\begin{equation}
\lambda < 1, \qquad 0<\Lambda \le 1. \tag{A3}
\end{equation}
If we suppose
\begin{equation}
G>\frac{2M}{\nu \lambda}\,, \tag{A4}
\end{equation}
which implies $\lambda/(2M)\ge 1/(n+1)$, the above inequality simplifies
to
\begin{multline}\label{4.3}
\left(\lambda-\frac1{2M}\right)\lambda\log\rho\ \le\
\frac{M+1}{2}\left(\frac{|l|}{\sigma}+\frac{1}{ \nu}\right) \\
\quad+\frac12\frac{\lambda^2}{ 1-\lambda}
\left[\log\left(\frac{\rho}{ 4\sigma\lambda}\right)+\frac32\right]
+{\frac{1}{2M}} \left[\log\left(\frac{2M\rho}{ \sigma}\right)+1\right].
\end{multline}
This inequality is obtained under assumptions (A1), (A2), (A3),
(A4) and the further assumption, implicitly made along the way, that $M$,
$\sigma A$ and $\nu G$ are integers.
We choose\footnote{\ We use here the ceiling function
$\lceil x\rceil = \min_{n\in\Z}\{n : n\ge x\}$.}
\begin{equation}\label{4.4}
\begin{split}
M =\lceil 2\lambda^{-2}\rceil.
\end{split}
\end{equation}
With this choice, (\ref{4.3}) can be replaced by
\begin{multline}\label{4.5}
\left(\lambda-\frac14\lambda^2\right)\lambda\log\rho \ \le\
\left(\frac{1}{ \lambda^2}+1\right)\left(\frac{|l|}{\sigma}+\frac{1}{ \nu}
\right) \\
+\frac12\frac{\lambda^2}{ 1-\lambda}
\left[\log\left(\frac{\rho}{ 4\sigma\lambda}\right)+\frac32\right]
+ \frac{\lambda^2}{ 4}
\left[\log\left(\frac{\rho}{ \sigma}
\frac{2+2\lambda^2}{\lambda^2}\right)+1\right].
\end{multline}
We now choose
\begin{equation}\label{4.6}
\nu=\frac{1}{ G}\,\left\lceil\frac{G\sigma}{|l|}\right\rceil,
\qquad \sigma=A^{-1}\left\lceil\frac{8|l|A}{
\lambda^4\log\rho}\right\rceil;
\end{equation}
note that $M$, $\sigma A$ and $\nu G$ are integers, hence our implicit
assumption is verified. An easy majorization of the right-hand side of
(\ref{4.5}) shows that
\begin{multline}\label{4.7}
\left(\lambda-\frac14\lambda^2\right)\lambda\log\rho \ \le\
\left(\frac{1}{ \lambda^2}+1\right)\frac14\lambda^4 \log\rho
+\frac{\lambda^2(3-\lambda)}{ 4(1-\lambda)}\log\rho \\
+\frac{\lambda^2}{ 4(1-\lambda)}
\left[\log\left(\frac{1+\lambda^2}{ 8 \sigma^3\lambda^4}\right)
+4\right].
\end{multline}
Since $\sigma\ge 8\lambda^{-4}(\log\rho)^{-1}$, we see that (\ref{4.7})
implies
\begin{multline}\label{4.8}
\left(\lambda-\frac14\lambda^2\right)\lambda\log\rho \ \le\
\left(\frac{1}{ \lambda^2}+1\right)\frac14\lambda^4 \log\rho
+\frac{\lambda^2(3-\lambda)}{ 4(1-\lambda)}\log\rho \\
+\frac{\lambda^2}{ 4(1-\lambda)}
\left[\log\left(8^{-4}(1+\lambda^2)\lambda^8(\log\rho)^3\right)+4 \right].
\end{multline}
Since $\lambda\log\rho\le 1$, inequality (\ref{4.8}) yields
\begin{multline}\label{4.9}
\left(\lambda-\frac14\lambda^2\right)\lambda\log\rho \ \le\
\left(\frac{1}{ \lambda^2}+1\right)\frac14\lambda^4 \log\rho
+\frac{\lambda^2(3-\lambda)}{ 4(1-\lambda)}\log\rho \\
+\frac{\lambda^2}{ 4(1-\lambda)}
\left[\log(\lambda^5+\lambda^7)-4.317 \right].
\end{multline}
Note that $\log(\lambda^5+\lambda^7)-4.317<\log 2 -4.317<-3.623<0$.
Dividing both sides of (\ref{4.9}) by $\lambda^2\log \rho$ and using the
lower bound $1/\log\rho \ge \lambda$ gives
\[
1-\frac14\lambda \le (1+\lambda^2)\frac14
+\frac{3-\lambda}{ 4(1-\lambda)} -3.623\,\frac{\lambda}{
4(1-\lambda)}
\]
and after multiplication by $4(1-\lambda)$ and an easy simplification we find
\[
0 \le - 0.623\lambda -\lambda^3 < 0.
\]
This is a contradiction, and shows that one of the hypotheses (A1) to
(A4), together with the choices (\ref{4.4}) and (\ref{4.6}), is
untenable. Therefore, (A1) does not hold if we assume (A2), (A3), (A4)
and (\ref{4.4}) and (\ref{4.6}). Our
choice
of parameters in (\ref{4.4}) and (\ref{4.6}) guarantees that (A2), (A3),
(A4) are verified, except possibly for the condition $s+1\le r$ in (A2) that
must be compatible with our choice of $\sigma$ in (\ref{4.6}). Let us
assume for the time being that this is the case. Then if
we assume the first half of (A1), namely $\log|1-\alpha^l|_v \le -\Lambda$,
we conclude that the second half of (A1) does not hold. Note
also that by (\ref{4.6}) we have
\begin{equation}\label{4.10}
\sigma \ge 8|l|\lambda^{-4}(\log\rho)^{-1} \quad\hbox{and}\quad
\nu \ge \sigma/|l| \ge 8\lambda^{-4}(\log\rho)^{-1};
\end{equation}
therefore, $2\lceil2\lambda^{-2}\rceil/(\nu\lambda) \ge \frac12
\lambda\log\rho$ and {\it a fortiori} (A4) can be replaced by $G\ge
\lambda\log\rho$.
If we recall that we had chosen $k=\lambda(s+1)(n+1)$, we conclude that
\begin{proposition}\label{Prop 1}
Let $K$, $v$, $r$, $a$, $\alpha=a^{1/r}$,
$\gamma$ be as
before. Assume that $A$, $\rho$, $G$, $\lambda$ satisfy
$r>A\ge h(a)$, $\rho=r/A$, $G\ge \max(h(\gamma),\lambda\log\rho)$ and
$0<\lambda <\min(1,1/\log\rho)$.
Suppose further that
\[
\log\big|1-\alpha^l\big|_v \le -\lambda\log\rho.
\]
Let
\[
\sigma = A^{-1}\left\lceil\frac{8|l|A}{ \lambda^4\log\rho}\right\rceil.
\]
Then if $\sigma \le \rho$ we have
\[
\log|1-\gamma^{-1}\alpha|_v >
-\lambda^2\,\left\lceil\frac{G\sigma}{|l|}\right\rceil\,
\left\lceil\frac{8|l|A}{ \lambda^4\log\rho}\right\rceil\,\log\rho.
\]
\end{proposition}
%%%%%%%%%%%%%%%%%%%%%%%%%%% SECTION 5 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{\kern-1mm Applications to diophantine approximation in a
number field by a finitely generated multiplicative group}
As a corollary of Proposition \ref{Prop 1}, we derive in this section
improvements of Theorem 1 and Theorem 2 of \cite{BoCo}. As in
that paper, we let $K(v)$ be the residue field of $K_v$ and
$f_v$, $e_v$ the residue class degree and ramification index
of the extension $K_v/\Q_v$. We abbreviate
\[
d^*_v=\frac{d}{ f_v\log p},\quad D^*_v=\max(1,d^*_v).
\]
We assume that $|a|_v=1$, so that if we choose $l=p^{f_v}-1$,
then $|a^l-1|_v<1$. {From} Lemma 1 of \cite{BoCo} we may suppose
that
\begin{equation}\label{5.1}
\begin{split}
\log\frac{1}{ |1-\alpha^l|_v}=\log\frac {1}{ |1-a^l|_v}\ge\frac{f_v\log
p}{ d}=\frac{1}{ d^*_v}\ge\frac{1}{ D^*_v}.
\end{split}
\end{equation}
Continuing with the notations of \S4, we suppose that
$r>2A$ and choose
\begin{equation}\label{5.2}
\begin{split}
\lambda=\left(D^*_v\log \rho\right)^{-1}.
\end{split}
\end{equation}
Then we can apply Proposition \ref{Prop 1} and deduce that
\begin{equation}\label{5.3}
\begin{split}
\log|1-\gamma^{-1}\alpha|_v >
-\lambda^2\,\left\lceil\frac{G\sigma}{|l|}\right\rceil\,
\left\lceil\frac{8|l|A}{ \lambda^4\log\rho}\right\rceil\,\log\rho
\end{split}
\end{equation}
provided that $G\ge\max(h(\gamma),{1/D_v^*})$ and also $\sigma\le\rho$.
With the modified height $h'(x)$ defined in the statement of the Main
Theorem,
the condition on $G$ becomes $G\ge h'(\gamma)$. Our choice for $A$ will be
$A=h'(a)$.
For the application we have in mind, $r$ must be relatively large compared
to $h'(a)$ if we want a nontrivial conclusion for our final result. Thus to
begin with we assume that
\begin{equation}\label{5.4}
\begin{split}
r> e^4 D_v^* h'(a).
\end{split}
\end{equation}
In particular, $\log\rho\ge 4$.
The next step in simplifying (\ref{5.3}) consists in removing the brackets in
the
ceiling function. By (\ref{4.10}), (\ref{5.4}), $A\ge 1/D_v^*$ and our choice
of $\lambda$ we have
\[
\left\lceil\frac{8|l|A}{ \lambda^4\log\rho}\right\rceil=A\sigma \ge 8 |l|
(D_v^*)^3 (\log\rho)^3 \ge 512,
\]
hence we may remove the brackets at the cost of multiplying by $1+1/512$,
at most. In a similar way, we have
\[
\left\lceil\frac{G\sigma}{|l|}\right\rceil\ge 8 G (D_v^*)^3(\log\rho)^3
\ge 512,
\]
because $G\ge 1/D_v^*$. Therefore, the cost of removing the brackets is
at most a factor of $1+1/512$. Again, removing the brackets from
$\sigma$ will not cost us more than a further factor $1+1/512$. Thus
the total cost in this simplification is at most a factor
$(1+1/512)^3<1.006$.
We can replace $h'(\gamma)$ by $h'(\gamma^{-1}\alpha)$, at a small cost.
Indeed, $h(\gamma)\le h(\gamma^{-1}\alpha)+h(\alpha)$, hence using $h'\ge
1/D_v^*$ we find
\begin{equation}\label{5.5}
\begin{split}
h'(\gamma) \le h'(\gamma^{-1}\alpha)+ \frac{h(a)}{ r}
< h'(\gamma^{-1}\alpha)+ \frac{1}{ e^4D_v^*}< (1+e^{-4})
h'(\gamma^{-1}\alpha).
\end{split}
\end{equation}
Thus the total cost of these simplifications is a factor of at most
$1.006\times (1+e^{-4}) < 1.03$. Therefore, after removing the brackets,
taking
into account this small correction and making a further rounding off of
constants, (\ref{5.3}) becomes the simpler
\begin{equation}\label{5.6}
\begin{split}
\log|1-\gamma^{-1}\alpha|_v > -66 p^{f_v}( D_v^*)^6 h'(a) \left(\log
\frac{r}{h'(a)}\right)^5 h'(\gamma^{-1}\alpha).
\end{split}
\end{equation}
This inequality has been obtained under the assumption that $s+1\le r$. If
however $s+1\ge r+1$, we must have
\[
\left\lceil \frac{8|l|A}{\lambda^4\log\rho}\right\rceil = A\sigma =s+1\ge
r+1=\rho
A+1,
\]
hence $8|l| \ge \lambda^4 \rho\log\rho$. With our choice of $\lambda$ and
$l$, this means that if
\begin{equation}
\rho(\log\rho)^{-3} \ge 8 p^{f_v} (D_v^*)^4\tag{A5}
\end{equation}
then the condition $\sigma\le \rho$ in Proposition 1 is verified.
We now summarize our results as follows.
%%%%%%%%%%%%%%%%%%%%%%%%%%%% THEOREM 1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem}\label{1}
Let $K$ be a number field of degree $d$ and $v$ an absolute value of $K$
dividing a rational prime $p$. Let $a\in K$ with $a$ not $0$ or a root of
unity, and suppose that $a$ satisfies $|a|_v=1$.
Let $r$ be a positive integer coprime to $p$. Then $a$ has an $r$th root
$\alpha\in K_v$ satisfying $0<|1-\alpha^{p^{f_v}-1}|_v<1$. Let
$\alpha'=\alpha\gamma^{-1}$ with $\gamma\in K$, $\gamma\ne0$. Let
$C=66\,\,p^{f_v}\,(D^*_v)^6$ and $0<\kappa$, and suppose that
\begin{equation}
r \ge \rho\left(\frac{C}{\kappa}\right) h'(a). \tag{H1}
\end{equation}
Then
\[
|\alpha'-1|_v\ge H'(\alpha')^{-r\kappa}.
\]
Moreover, if\/ $|a-1|_v<1$ then $a$ has an $r$th root $\alpha\in K_v$
satisfying
\[
0<|\alpha-1|_v<1,
\]
and (H1) can be further improved by replacing $C$ with the smaller
constant $C'=66 (D_v^*)^6$.
\end{theorem}
\begin{remark}\label{Rem}\rm Before completing the proof of
Theorem~\ref{1}, a comparison with the explicit result in \cite{Bu} is
in order. To avoid undue complications, we only consider asymptotic bounds
as $h(\gamma) \to \infty$ and $r/h'(a)\to \infty$. Then, with the
optimal choice of $\kappa$, the bound given
by our Theorem~\ref{1} is
\[
\log\frac{1}{|\alpha'-1|_v} \ \le\
\bigl(66+o(1)\bigr)\, p^{f_v}\, (D^*_v)^6 h'(a)\left(\log\frac
{r}{h'(a)}\right)^5 h(\gamma).
\]
On the other hand, from \cite{Bu} we may show that
\[
\log\frac{1}{|\alpha'-1|_v} \ \le\ \bigl(24+o(1)\bigr)\, p^{f_v}\,
(d^*_v)^4 h'(a)\left(\log\frac {r}{h'(a)}\right)^2 h(\gamma),
\]
which is better than Theorem~\ref{1}.
Thus the interest of Theorem~\ref{1} is more in the method of proof than
in the result itself.
\end{remark}
\begin{proof} By (\ref{5.6}) it suffices that $r$ be so big that
\[
\kappa r \ge C h'(a) \left(\log\frac{r}{ h'(a)}\right)^5,
\]
that is $\rho (\log \rho)^{-5} \ge C/\kappa$. Note that, with our value for
$C$, this condition takes care of (A5) as soon as $\kappa \le 8 (D_v^*)^2$.
On the other hand, we have the Liouville lower bound
\[
|\alpha'-1|_v\ge (2H'(\alpha'))^{-r},
\]
while $H'(\alpha')^{D_v^*}\ge e >2$,
hence in any case we have $|\alpha'-1|_v\ge H'(\alpha')^{-2D_v^*r}$. This
shows that the conclusion of Theorem~\ref{1} is trivial as soon as
$\kappa > 2D_v^*$. Thus condition (A5) is of no consequence for the
verification of Theorem~\ref{1}, completing the proof.
\end{proof}
In applications, condition (H1) is the most important. A direct
comparison with Theorem 1 of \cite{BoCo} shows a big improvement
in the absolute constant of (H1) and a reduction in the power of
the logarithmic term from $7$ to $5$. The condition (H2) of \cite{BoCo}
is now eliminated.
%%%%%%%%%%%%%%%%%%%%%%%%%%%% THEOREM 2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem}\label{2}
Let $K$ be a number field of degree $d$ and $v$ a place of $K$
dividing a rational prime $p$.
Let $\Gamma$ be a finitely generated subgroup of $K^*$ and let
$\xi_1,\dots,\xi_t$ be generators of $\Gamma/\mathrm{tors}$. Let
$\xi\in\Gamma$, $A\in K^*$ and $\kappa>0$ be such that
\[
0<|1-A\xi|_v t D_v^*$.
We need to bound $h'(a)$ and for this we use (\ref{5.7}). In view of
(\ref{5.11}), $r\ge 4$ and $h(\xi)\le h(A\xi)+h'(A)$, we have
\begin{equation}\label{5.12}
\begin{split}
h'(a) \le h'(A)+ \frac{1}{2\rho}\, r + \frac 4 r \,h(\xi)
\le 2\,h'(A)+\frac{1}{2\rho}\, r + \frac 4 r \,h(A\xi) .
\end{split}
\end{equation}
Now we choose $N$ to be
\[
N =\left\lceil 2(p^{f_v}-1)Q\left(1+\max\left(8\rho h'(A),\sqrt{16\rho
h(A\xi)}\right)\right)\right\rceil.
\]
Then (\ref{5.8}) implies that
\[
r\ge \max\left(8\rho h'(A),\sqrt{16\rho h(A\xi)}\right)
\]
hence (\ref{5.12}) yields
\[
h'(a) \le \frac{1}{ 4\rho}\, r +\frac{1}{2\rho}\, r + \frac{1}{ 4\rho}\,
r
=\frac {1}{\rho}\,r,
\]
hence (\ref{5.9}), and {\it a fortiori} (\ref{5.10}), holds with this choice
of $N$.
On the other hand, $r\le N+3$ and finally from (\ref{5.10}) we have
\[
h(A\xi) \le (D_v^*)^{-1}\left\lceil 2(p^{f_v}-1)Q\left(1+\max\left(8\rho
h'(A),\sqrt{16\rho h(A\xi)}\right)\right)\right\rceil + 3.
\]
The first alternative for the maximum easily yields
\[
h(A\xi) \le 16 p^{f_v}\rho h'(A) Q,
\]
because $\rho h'(A) Q$ is fairly large (use $\rho\ge 33 D_v^*$ to get
$\rho h'(A) Q\ge (66 t)^t$), hence the small corrections in
going from $1+\max$ to $\max$ and
in removing the ceiling brackets and the constant $3$ are easily absorbed in
replacing $p^{f_v}-1$ by $p^{f_v}$.
The second alternative for the maximum yields
\[
h(A\xi) \le 2 p^{f_v} Q \sqrt{16\rho h(A\xi)}
\]
and finally
\[
h(A\xi) \le 64 p^{2f_v} \rho Q^2,
\]
completing the proof of Theorem~\ref{2}.\end{proof}
\section{Appendix: from a private communication by David
Masser}
In this appendix, we reproduce material from a letter of David Masser to
the first author dated 8th January 1984. These ideas of Masser inspired
our \S2 and are reproduced here with his permission.
``\dots My own method was based on zero estimates rather than heights,
using a `dividing out' trick from transcendence. It gives the
following general result.
\begin{Mthm} Suppose $\theta$ is algebraic of
degree $d\ge 2$ and of absolute height $H\ge1$. Fix an integer $e$
with
\[
1\le e1.
\]
Then the effective strict type of $\theta$ is at most
\[
\frac{-e\gamma\log|\theta-\frac{p_0}{q_0}|}{\log\Lambda}\,.
\]
\end{Mthm}
I didn't try to improve the constant $4$, although this could
certainly be done by using asymptotics for binomial coefficients.
The proof can be expressed in three lemmas, where
$c_1,c_2,\dots$ denote constants depending only on $d$, $H$,
$\ep$. For $P(x,y)$ in ${\C}[x,y]$ write as in
Siegel's set-up
\[
P_l(x,y)=\frac{1}{l!}\left(\frac{\partial\;}{\partial
x}\right)^lP(x,y).
\]
\begin{Mlem} For each $k\ge1$ there exists a
nonzero polynomial $P(x,y)$ in ${\Z}[x,y]$, of degree at
most $\delta k$ in $x$ and at most $e$ in $y$, with coefficients
of absolute value at most $c_1(4H)^{\alpha k}$, such that
\[
P_l(\theta, \theta)=0,\qquad (0\le l\frac{d}{e+1}+e
\]
as in Siegel. The optimal choice $e=10$,
$\ep=\frac{\sqrt2-1}{11}$ gives any exponent
\[
\lambda>\frac{55}{14}\left(4+\sqrt2\right)=21.270\dots
\]
for the real root $\theta(m,d)$ of $x^d-mx^{d-1}+1=0$ provided
$d\ge d_0(\lambda)$ and $m\ge m_0(d)$.
I briefly looked at a similar approach in the Gelfond--Dyson
set-up, with a fixed integer $t$ and derivatives
$(\partial/\partial x)^l(\partial/\partial y)^sP(x,y)$ for
\[
\frac lk+\frac st<1
\]
But even if the analogous zero estimate could be made to work, it
seems as if $t=1$ (i.e.\ Siegel) gives the best results for
$\theta(m,d)$. So I didn't try too hard with this.''
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\bigskip
\noindent
Enrico Bombieri, \\
School of Mathematics, \\
Institute for Advanced Study, \\
Princeton, N.J. 08540, USA\\
e-mail: eb@math.ias.edu
\medskip
\noindent
Paula B. Cohen, \\
Department of Mathematics, \\
Texas A\&M University, \\
College Station, TX 77843, USA\\
e-mail: pcohen@math.tamu.edu
\end{document}