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 \(m = \lfloor \log^+ N_{\le\lambda}(F) \rfloor\).
Let \(|f|_\lambda := \sup_i |a_i|p^{\lambda i} = |f(x/\alpha_\lambda)|_{\text{Gauss}}\), where \(v(\alpha_\lambda)=\lambda\) and \(|\alpha_\lambda| = p^{-\lambda}\). So we can view \(|f|_\lambda\) as the maximum modules \(\sup_{|x|\le p^\lambda} |f(x)|\) on the closed disc \(D(0,p^\lambda)\) of radius \(p^\lambda\).
We can prove that
\(|f|_\lambda \le 1 \Leftrightarrow \mathrm{Slope}_{\ge \lambda}(f)\).
On polynomials, \(|fg|_\lambda = |f|_\lambda |g|_\lambda\) since Gauss norm is multiplicative.
\(|f+g|_\lambda \le \max(|f|_\lambda, |g|_\lambda)\), it is a non-archimedean norm.
Equivalent formulation of theorem using norm:
If all coefficients of the monic differential operator \(\mathcal{P}\) satisfy \(|a_i|_\delta \le 1\) and \(\lambda < \delta_- - \frac{1}{p-1}\), then
\[ N_{\le \lambda}(F) \le N_{<\delta}(\mathcal{P}F) + \frac{m}{-\lambda + \delta_- - \frac{1}{p-1}} + \deg \mathcal{P}.\]
Or, in terms of zeros in a disc (closed) or ball (open),
\[ N_{D(0,p^\lambda)}(F) \le N_{B(0,p^\delta)}(\mathcal{P}F) + \frac{m}{-\lambda + \delta_- - \frac{1}{p-1}} + \deg \mathcal{P}.\]
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.
It is also possible that we can use this in conjunction with some concrete differential operators, to produce an algorithm that will bound the number of zeros of a generic section.
After some thought I found that the norm
\[ |f|_\lambda = \sup_i |a_i|p^{\lambda i} \]
seems to almost match what I previously described the slope constraint
\[ \mathrm{Slope}^\mu_{\ge\lambda}(f) := \forall i\ge 0, v(a_i) \ge \lambda i + \mu. \]
I want to discuss how to generalize the result, so that we can make use of the values of various \(|a_i|_\delta\) to bound the number of zeros of a solution of a differential equation, instead of just using \(|a_i|_\delta\le 1\).
Make the following definitions
\[ d_\lambda(f) = \inf_{i\ge 0} \{ v(a_i) - \lambda i \}. \]
Then we have the equivalence relations
\[ |f|_\lambda \le p^{-\mu} \Leftrightarrow d_\lambda(f) \ge \mu \Leftrightarrow \mathrm{Slope}_{\ge\lambda}^\mu(f). \]
Let \(\mathrm{NP}(f)\) be the Newton polygon of \(f\), then draw a line of slope \(\lambda\) and tangent to \(\mathrm{NP}(f)\). This line will intersect with \(y\) axis at \(\mu = d_\lambda(f)\), and the Gauss norm will be \(|f|_\lambda = p^{-\mu}\).
This enable us to understand the norm easily.
Example. \(f = 1+t^3\), then
\[|f|_\lambda = \begin{cases} p^{3\lambda} & \lambda > 0, \\ 1 & \lambda \le 0. \end{cases} \]
If a series \(f=\sum a_i t^i\) has convergence radius \(r=p^\sigma\), then
\[|f|_\lambda = \begin{cases} \infty & \lambda > \sigma, \\ \text{depends on convergence of $f$ at $|t|=p^\lambda$} & \lambda = \sigma, \\ <\infty & \lambda < \sigma. \\ \end{cases} \]
Consider \(f = \frac{t^a}{1-t}=t^a(1+t+t^2+\dots)\), then
\[|f|_\lambda = \begin{cases} \infty & \lambda > 0, \\ p^{a\lambda} & \lambda \le 0. \end{cases} \]
We can prove that the norm is multiplicative on \(\mathbb{Q}[t]\) as well as \(\mathbb{Q}_p[t]\), but I’m not sure if it is multiplicative on \(k[\![t]\!]\) for extensions \(k/\mathbb{Q}_p\), or for \(k=\mathbb{C}_p\). Definitely not multiplicative on rational functions as \(1/(1-t)\cdot (1-t)\) shows a conterexample. It is sub-multiplicative.