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Thue: $P(X, Y) = R_1(X) + Y R_2(X)$.

Let $L$ be a linear form in $K[X_1, \ldots, X_N]$
where $K$ is a number field.

For $v \in M_K$: $|L|_v = \max(|a_j|_v)$ where $L
= a_1 X_1 + \cdots + a_N X_N$. Then define
\[
  H(L) = \left(\prod_{v \in M_K}
  |L|_v^{d_v}\right)^{\frac{1}{[K : \Qbb]}}
.\]
By the product formula, this is invariant under
multiplication by an element $\alpha \in
K^\times$:
\[
  |\alpha L|_v = |\alpha|_v |L|_v
,\]
so
\[
  H(\alpha L) = \prod_{v \in M_K} |\alpha
  L|_v^{d_v} = H(L) \prod_{v \in M_K}
  |\alpha|_v^{d_v} = H(L)
.\]

\begin{lemma*}[Siegel's lemma]
  \label{siegel}
  Let $K$ be a number field of degree $D$. Let $M,
  N \in \Zbb_{> 0}$ such that $N > MD$ and let $\mathcal{H}
  \in \Rbb_{\ge 1}$. Let $L_1, \ldots, L_M \in
  K[X_1, \ldots, X_N]$ be linear forms such that
  $H(L_j) \le \mathcal{H}$. Then there exist $x_1,
  \ldots, x_N \in \Zbb$ (not all $0$) such that $L_h(x_1,
  \ldots, x_N) = 0$ for $j = 1, \ldots, M$ and
  \[
    |x_i| \le (N \mathcal{H})^{\frac{MD}{N - MD}}
  .\]
\end{lemma*}

In particular, if $N \ge MD$, then the bound is
$N \mathcal{H}$.

There is a refinement of this lemma which is due
to Bombieri and Vaaler.

\begin{corollary*}
  Let $\alpha$ be an algebraic number of degree
  $D$. Let $w_1, w_2, \delta \in \Rbb_{> 0}$, and let $I
  \in \Rbb_{> 0}$. Let $n_1, n_2 \in \Zbb_{> 0}$.
  Suppose that
  \[
    |\{(i_1, i_2) \in \Zbb_{\ge 0}^2 : i_1 w_1 +
    i_2 w_2 < I\}| \le \frac{(n_1 + 1)(n_2 +
    1)}{(1 + \delta) D}
  .\]
  Then there exists $P \neq 0 \in \Zbb[X_1, X_2]$
  of degree $n_j$ in $X_j$ such that $I_P(\alpha,
  \alpha, w_1, w_2) \ge I$ and
  \[
    H(P) \le (4H(\alpha))^{(n_1 + n_2)\delta^{-1}}
  \]
  where $H(P)$ is the maximal absolute value of
  hte coefficients.
\end{corollary*}

\begin{proof}
  For $(i_1, i_2)$ consider:
  \[
    L_{i_1, i_2} = \sum_{j_1 = 0}^{n_1} \sum_{j_2 =
    0}^{n_2} {j_1 \choose i_1} {j_2 \choose i_2}
    a_{j_1, j_2} \cdot \alpha^{j_1 - i_1 + j_2 -
    i_2}
  \]
  where $a_{j_1, j_2}$ are variables of $L_{i_1,
  i_2}$.
  Then
  \[
    L_{i_1, i_2}((a_{j_1, j_2})_{j_1, j_2}) = 0
    \iff P_{i_1, i_2}(\alpha, \alpha) = 0
  \]
  where
  \[
    P = \sum_{j_1 = 0}^{n_1} \sum_{j_2 = 0}^{n_2}
    a_{j_1, j_2} X_1^{j_1} X_2^{j_2}
  .\]
  Need to find $(a_{j_1, j_2})_{j_1, j_2}$ such
  that $L_{i_1, i_2}((a_{j_1, j_2})) = 0$ for all
  $i_1, i_2$ with $i_1 w_1 + i_2 w_2 \le I$.

  Apply \nameref{siegel}:
  \[
    N = (n_1 + 1)(n_2 + 1), \qquad M \le
    \frac{N}{(1 + \delta) D}
  .\]
  Then
  \[
    \frac{MD}{N - MD} \le \frac{MD}{(1 + \delta)
    MD - MD} = \delta^{-1}
  .\]
  We need to estimate $H(L_{i_1, i_2})$. For
  finite places $v$,
  \[
    |L_{i_1, i_2}|_v \le \max(1, |\alpha|_v)^{n_1
    + n_2}
  .\]
  For infinite places:
  \[
    |L_{i_1, i_2}|_v \le 2^{n_1} \cdot 2^{n_2}
    \max(1, |\alpha|_v)^{n_1 + n_2}
  \]
  Then
  \[
    H(L_{i_1, i_2}) \le 2^{n_1 + n_2} \cdot
    H(\alpha)^{n_1 + n_2} \eqdef \mathcal{H}
  .\]
  Then \nameref{siegel} gives the bound
  \[
    [2^{n_1 + n_2} H(\alpha)^{n_1 + n_2} \ub{(n_1 +
    1)(n_2 + 1)}_{\le 2^{n_1 + n_2}}]^{\delta^{-1}}
    .
    \qedhere
  \]
\end{proof}

\begin{proof}[Proof of \nameref{siegel} for $K =
  \Qbb$]
  We can assume that the coefficients of each
  $L_j$ are integers, and that they are relatively
  prime. Then each coefficient is bounded by
  $\mathcal{H}$. Take $Y = \left\lfloor (N
  \mathcal{H})^{\frac{MD}{N - MD}} \right\rfloor$.

  Consider $(y_1, \ldots, y_N) \in \{0, 1, \ldots,
  Y\}^N$. Evaluating $L_j$ at all such $(y_1,
  \ldots, y_N)$ we have
  \[
    \max L_j(y_1, \ldots, y_N) - \min L_j(y_1,
    \ldots, y_N) \le Y \cdot \mathcal{H} N
  .\]
  The number of possible values of $L_j(y_1,
  \ldots, y_N)$ is $\le Y \cdot H \cdot N + 1$.

  \textbf{Claim:} $(Y \mathcal{H} \cdot N + 1)^M <
  (Y + 1)^N$.

  Indeed:
  \begin{align*}
    Y
    &= \left\lfloor (N\cdot
    \mathcal{H})^{\frac{M}{N - M}} \right\rfloor
    \\
    Y + 1 > (N \cdot \mathcal{H})^{\frac{M}{N -
    M}} \\
    (Y + 1)^N > (N \cdot \mathcal{H})^M \cdot (Y +
    1)^M
  \end{align*}
  The claim follows by
  \[
    N \mathcal{H} Y + 1 < N \mathcal{H}(Y + 1)
  .\]
  Note that the above line uses the fact that
  $\mathcal{H} \ge 1$!

  By the box principle, there exist $(y_1, \ldots,
  y_N) \neq (z_1, \ldots, z_N)$, with entries
  bounded by $Y$, such that
  \[
    L_j(y_1, \ldots, y_N) = L_j(z_1, \ldots, z_N)
    \qquad \forall j = 1, \ldots, M
    .
    \qedhere
  \]
\end{proof}