Quasi unipotent monodromy
\(X\supset Y\), \(X-Y=D\) is a SNC divisor, quasi unipotent monodromy means we can talk about the action, you have a vector bundle \((\mathcal{V}, \nabla)\) on \(Y\), it corresponds to a local system \(V\) with action of \(\pi_1(Y_\mathbb{C}) \xleftarrow{} \mathrm{Gal}(\mathbb{C}(Y))\supset I_v\) the inertia subgroup, \(v\) an irrediciblt char of \(D\).
The Problem
When we are given a flat connection \(\mathcal{V}\) and we write it in local chart on an affine open \(U\), in coordinate form we have something like \(d - \Lambda\), where \(\Lambda\in M_n(\mathcal{O}(U))\).
Over a point \(x\) outside \(U\), we can have a local chart on which \(\mathcal{V}\) is isomorphic to trivial connection. This means we can have an isomorphism \(J\) that sends
\[ \Lambda \mapsto J^{-1} \Lambda J + J^{-1} d J = 0 \]
so that the equation we get is \(\Lambda J + d J = 0\). Solving it is equivalent to finding \(\Lambda = - dJ J^{-1} = - dJ J^* / \det J\).
What we want to do is to bound the degree of the rational functions inside \(\Lambda\), and this amounts to bound the degree of poles of entries of \(\Lambda\), and thus amounts to bounding the degree of poles of entries of \(J\) and the degree of zero of \(\det J\).
The Pieces Of The Puzzle
We have developed some magic that will bound the number of zeros of a solution \(F\) of a \(p\)-adic differential equation using the degree and the shape of the differential operator \(\mathcal{D}\) and the information (newton polygon or number of zeros / poles) of the differentiated function \(\mathcal{D}F\). To use this magic, it is necessary to know the differentiated function \(\mathcal{D}F\), as well as the degrees of the differential operator \(\deg \mathcal{D}\). There are also some restriction on the coefficients. Notably we must require \(v_p(a_i[t^0])\ge 0\) for this method to work.
If we want to apply the magic to bound the zeros of a flat section of a flat connection, whose local equation looks like \(d - \Lambda\), we need to use some ways to produce a differential equation. (Here we have a hole to be filled here)
This differential equation will have its coefficients generated by the entries of \(\Lambda\) and its derivatives. So it suffices to have bounds on the degrees of the entries of \(\Lambda\) and the derivatives of the entries of \(\Lambda\).
Now we are at the problem of bounding \(\Lambda\), which as we have seen can be converted to a problem of bounding \(J\) at each local point outside \(U\). Here we will need a geometric and intrinsic argument that will relate global quantities about the connection to bounds on poles of \(J\) and zeros of \(\det J\).
So in summary, the adventure map looks like this:
It might prove to be useful if we can find an example and understand what exactly we need to bound in the last step.
Step 1. Global Invariants To Local Bounds on \(J\)
This step is still mysterious by now. We think that global invariants like degree of vector bundles, which is the degree of its top line bundle, the wedged bundle, can control the degree of poles of \(J\) and zeros of \(\det J\).
Possibilities: Look at the stability formalisms, HN filtrations and polygons.
Step 2. Local Bounds on \(J\) To Bounds on \(\Lambda\)
This should be straightforward once we notice the relation \(\Lambda = dJ \cdot J^*/\det J\). Whose entries are products of entries of \(J\) and \(dJ\). If we have that \(\deg J\) and \(\deg dJ\) are bounded, then we can bound the degree of \(\Lambda\) as
\[ \deg \Lambda \le \deg dJ + (n-1)\deg J + \deg (\det J) \]
where \(n\) is the rank of the connection.
Step 3. Bounds on \(\Lambda\) To Bounds on \(\deg\mathcal{P}\) and the coefficients of \(\mathcal{P}\)
This is still mysterious to me but should be approachable. The key point is to convert a connection matrix to the differential equation of the flat section.
either we use something similar to a cyclic vector theorem and convert it this way.
or we just continue differentiate, use some other ways to obtain the linear dependence
Step 4. Bounds on \(\deg\mathcal{P}\) and the coefficients of \(\mathcal{P}\) To Bounds on zeros of flat sections
This is the magic we have developed in the previous adventure. But I think it is not clear whether the conditions can be met and we might need to enhance the magic.