Let me summarize the results I got on \(p\)-adic differential equations.
Consider the ring \(k[\![x]\!]\) of formal power series in \(x\) with coefficients in a field \(k=\mathbb{Q}_p\). Let \(D=\frac{\partial }{\partial x}\) be the derivation with respect to \(x\) on \(k[\![x]\!]\) defined by \(D(x^n)=nx^{n-1}\) for \(n\geq 0\).
We make the following convention
Denote \(\log_p^+ n = 1_{n\ge 1} \log_p n\), here \(p\) always denotes a prime number.
We use \(\mathcal{P}\) to denote a \(p\)-adic differential operator of shape \[\mathcal{P}=\sum_{i\le d} a_i(x)D^i = \sum_{i\le d} a_{ij}x^jD^i\] where \(a_i(x)\in k[\![x]\!]\) and \(a_{ij}\in k\). We say \(\mathcal{P}\) is integral if \(a_i(x)\in k[\![x]\!]\) for all \(i\), and monic if \(a_d(x)\in \mathbb{Z}_p[\![x]\!]^\times\), this is the same as assuming \(v(a_{d0})=1\). \(d\) is called the degree of \(\mathcal{P}\), we denote it by \(\deg\mathcal{P}\).
Let \(F\in k[\![x]\!]\) be a formal series and \(N_{\le\lambda}(F)\) denote the total length of slopes of the Newton polygon of \(F\) that lies in \([-\infty,\lambda]\).
Here we make the convention that only the part of \(F\) starting from degree \(0\) to the first non-zero term is considered as length of slope \(-\infty\), for example a zero function should have \(0\) length of slope \(-\infty\), i.e. we define \(N_{\le\lambda}(0)=0\) for any \(\lambda\).
For a series \(F=\sum_{i\ge 0} a_i x^i\in k[\![x]\!]\) and any real number \(\lambda\), define a constraint called \(\mathrm{Slope}_{>\lambda}(F)\), which means \[v(a_{0})\ge 0,\quad v(a_{i})>\lambda i\quad \forall i> 0.\] Similarly define \(\mathrm{Slope}_{\ge \lambda}(F)\) by replacing the strict inequality with non-strict inequality. We say that \(\mathrm{Slope}_{\ge \lambda}(\mathcal{P})\) is satisfied if \(\mathrm{Slope}_{\ge \lambda}(a_i(x))\) is satisfied for all \(i\).
It seems that the set \(S_{>^*\delta} := \{F\in k[\![x]\!]:\mathrm{Slope}_{>^*\delta}(F)\}\) form an \(\mathbb{Z}_p\)-algebra.
Proof. Let \(F,G\in S_{>^*\delta}\), we try to prove that \(F+G\) and \(FG\) are in \(S_{>^*\delta}\).
\(v(a_0+b_0)\ge \max(v(a_0),v(b_0))\ge 0\).
\(v(a_i+b_i)\ge \max(v(a_i),v(b_i))>^* \delta i\) for \(i>0\).
\(v\left(\sum_{i+j=n} a_ib_j\right) \ge \max_{i+j=n}(v(a_i)+v(b_j))>^* \max_{i+j=n}(\delta i+\delta j) = \delta n\).
Moreover it is a differential subalgebra because under differentiation the valuations can only increase.
The following is our result on the estimates of the number of solutions \(F\) of a \(p\)-adic differential equation \(\mathcal{P}\) in terms of the slopes of \(\mathcal{P}F\), degree of the operator and the shapes of the coefficients of \(\mathcal{P}\).
Theorem. Let \(\mathcal{P}\) be a monic differential operator and \(\lambda,\delta\) be real numbers, if we have \(\mathrm{Slope}_{>^*\delta}(\mathcal{P})\), and \(\lambda < \delta_- - \frac{1}{p-1}\), then
\[ N_{\le \lambda}(F) \le N_{\le^*\delta}(\mathcal{P}F) + \frac{m}{-\lambda + \delta_- - \frac{1}{p-1}} + \deg \mathcal{P}.\]
Here \(p\) is a prime, \(\delta_- = \min(0, \delta)\) is the negative part and \(m = \lfloor \log^+ N_{\le\lambda}(F) \rfloor\).
This follows from a general geometric method and can potentially be improved by tailoring the method to the specific situation. The next things we are looking into are
Find applications and understand what kind of conditions we need and what are needed to be bounded.
Remove or loosen the restriction on the slope of coefficients of \(\mathcal{P}\). This may involve either
allowing \(v(a_i[t^0])\) to be negative, or
equivalently, putting a positive valued coefficients on the head of \(\mathcal{P}\).
removing the monic requirement, try to find a much more general statement, perhaps involve a new concept of a newton polygon of a whole differential operator
In practical applications we would require \(F\) to be convergent on some ball of radius \(p^\lambda\) so \(N_{\le\lambda}(F)\) is guaranteed to be finite.
In combination of the differential algebra structure of the slopes, if we have a differential operator with crazy coefficients coming from \(\mathbb{Z}_p\)-polynomials of \(F_i\) adn its derivatives, if we can make sure the head coefficient is monic and \(F_i\) all in \(S_{>^*\delta}\), then we can enjoy the estimate.
I think the constraint \(a_0\in \mathbb{Z}_p[\![t]\!]^\times\) can be relaxed into \(v(a_{00})=0\), the other coefficients need not be integral.