\section{Three inputs from Fischler}\label{sec:inputs}

We first state the criterion that turns linear forms into a dimension
bound. The other two inputs provide the rational functions from which
the forms are built and the multiplicity estimate used to prove their
rank condition.

\subsection{A refined linear-independence criterion}

The rows of the matrix below are the coefficients of integer linear
forms in the target numbers. The two size bounds measure the growth of
those coefficients and the decay of the forms when evaluated at the
target vector. The rank condition is the third hypothesis to be proved.

\begin{proposition}[{\cite[Proposition~1]{F26}, over $\Q$}]
\label{prop:criterion}
Let $\theta=(\theta_0,\ldots,\theta_v)^T$ be a real vector with
$\theta_0\neq0$.
For each $n$ in an infinite set, let $L_n$ be an integer matrix with
$v+1$ columns and any finite number of rows.
Suppose that, for fixed $B>1$ and $\alpha<1$ with $\alpha>0$,
\[
 \max_{i,j}|(L_n)_{i,j}|\leq B^{n+o(n)},\qquad
 \max_i|(L_n\theta)_i|\leq\alpha^{n+o(n)}.
\]
Suppose also that, for sufficiently large $n$,
\begin{equation}\label{eq:weak-rank}
 L_nx=0,\quad x\in\Q^{v+1}\quad\Longrightarrow\quad x_0=0.
\end{equation}
Then
\[
 \dim_\Q\Span_\Q\{\theta_0,\ldots,\theta_v\}
       \geq1-\frac{\log\alpha}{\log B}.
\]
\end{proposition}

Condition \eqref{eq:weak-rank} says that the first column is not a
rational linear combination of the other columns. Full column rank is
not required. This distinction is useful here because the rational
function in Proposition~\ref{prop:rational} is obtained by an existence
argument, and its coefficients need not give full rank.
In our application, $\theta_0=1$ and the other coordinates are even beta
values. We will bound the coefficients and values of the forms in
Section~\ref{sec:estimates}, then prove \eqref{eq:weak-rank} in
Section~\ref{sec:rank}. No lower bound on the value of each individual
form is required.

\subsection{A rational function with small coefficients}

The construction supplies two properties at once: integer
partial-fraction coefficients of controlled size, and high-order
vanishing at infinity. The latter means that the function decays rapidly
at the arguments where we will sum it.

\begin{proposition}[{\cite[Theorem~4]{F26}}]\label{prop:rational}
Let $a$ be a positive integer and let $\omega,\Omega,r$ be rational
numbers with
\[
 a>\Omega\geq\omega\geq1,\qquad r\geq1.
\]
For every sufficiently large integer $n$ such that $rn,\omega n,\Omega n$
are integers, there are integers $c_{i,j}$, not all zero, for
$1\leq i\leq a$ and $0\leq j\leq n$, with the following properties.
Set
\[
 F_n(t)=\sum_{i=1}^a\sum_{j=0}^n\frac{c_{i,j}}{(t+j)^i}
       =\sum_{\nu\geq1}A_\nu t^{-\nu}.
\]
Then $A_\nu=0$ for $\nu<\omega n$,
\begin{equation}\label{eq:coeff-small}
 |c_{i,j}|\leq \exp(n\log\chi+o(n)),
 \quad
 \log\chi=
 \frac{\omega\log2+3\omega^2+\omega^2\log(a+1)
                +\frac12\Omega^2\log r}{a-\omega},
\end{equation}
and, uniformly in $\nu$,
\begin{equation}\label{eq:laurent-bounds}
 |A_\nu|\leq \exp(n\log\chi+o(n))\,n^\nu\nu^a
 \begin{cases}
 r^{\nu-\Omega n},&\nu<\Omega n,\\
 1,&\nu\geq\Omega n.
 \end{cases}
\end{equation}
\end{proposition}

Here $a$ bounds the pole orders, and $\omega n$ is a lower bound for
the order of vanishing at infinity. The parameter $r$ will determine
the summation cutoff $rn$.
The larger threshold $\Omega n$ gives the stronger estimate in the first
line of \eqref{eq:laurent-bounds}. These parameters are fixed as $n$ grows;
their choice is postponed until Section~\ref{sec:conclusion}.

Group the coefficients with the same pole order into polynomials
\[
 P_i(z)=\sum_{j=0}^n c_{i,j}z^j,
 \qquad b=\max\{i:P_i\not\equiv0\}.
\]
The $P_i$ will multiply the special functions obtained by summing $F_n$.
We retain the actual largest index $b$, rather than replacing it by $a$,
because the rank proof needs a nonzero highest coefficient $P_b$.
The integer $b$ may depend on $n$, but always satisfies $1\leq b\leq a$.

\subsection{A multiplicity estimate}

The rank proof will argue by contradiction. If the coefficient matrix
has a forbidden kernel vector, we can construct polynomial combinations
of solutions with many vanishing derivatives. The following theorem
limits their total order of vanishing in terms of the polynomial degrees.
It plays the role of a zero-counting bound for these combinations.

Let $Y'=\matA(w)Y$ be a fixed rational differential system of size $q$.
For polynomials $S_0,\ldots,S_{q-1}$ of degree at most $m$, write
\[
 \mathcal R(Y)=\sum_{j=0}^{q-1}S_jY_j.
\]
We call this scalar function the \emph{remainder} associated with the
solution $Y$ and the fixed polynomial row $(S_0,\ldots,S_{q-1})$.
The selected remainders must satisfy a common scalar linear differential
equation of some order $\mu$, with rational coefficients. In our
application such an equation will follow from the finite dimension of
the system.

At a point $\sigma$, the Nilsson class consists of finite sums of
holomorphic functions multiplied by complex powers and nonnegative
integer powers of a logarithm of the local coordinate.
At infinity the local coordinate is $1/w$.
The order is the least real part of a power that occurs after expanding
the holomorphic factors; logarithmic factors do not change it. For example,
$\ord_0(w^e(\log w)^j)=\operatorname{Re}e$.
In particular, $w^N$ has order $-N$ at infinity. Negative orders will
therefore occur when we apply the theorem.

\begin{proposition}[{\cite[Theorem~3]{F26};
see also \cite[Theorem~3.1]{F18}}]\label{prop:multiplicity}
Let $\Sigma$ be a fixed finite set of points of the Riemann sphere
containing infinity. At each $\sigma\in\Sigma$, choose a family of local
solutions $Y_{\sigma,j}$, indexed by $j\in J_\sigma$, whose remainders
are in the Nilsson class and are linearly independent over $\C$.
If a nonzero rational differential operator of order $\mu$ annihilates
all these remainders, then
\[
 \sum_{\sigma\in\Sigma}\sum_{j\in J_\sigma}
       \ord_\sigma\mathcal R(Y_{\sigma,j})
       \leq(m+1)(\mu-|J_\infty|)+C.
\]
The constant $C$ depends only on the differential system and on $\Sigma$.
\end{proposition}

The solution families at different points need not agree by analytic
continuation, and no independence between different families is required.
These features are explicit in \cite[Section~3.1]{F18}.
We use the formulation in \cite{F26}, which also allows negative orders
at infinity, as in its proof of Proposition~3.
The constant is uniform in the polynomial row and in the selected
solutions. This uniformity is essential when that row and the solutions
depend on $n$.