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Suppose $K$ is a number field, and $A\in M_n(K)$. $v$ is a place of $K$, and $f_1,\cdots,f_n$ are analytic functions (one variable) on $m_v\mathcal O_{K,v}$, satisfying: $\frac{\mathrm d \bf {f}}{\mathrm d t}=A\bf {f}$ where ${\bf {f}}=(f_1,\cdots,f_n)^T$, and the initial value ${\bf f}(0)=(c_1,\cdots,c_n)$ is known. Suppose there is a polynomial $g(x_1,\cdots,x_n)\in \mathcal O_{K_v}[x_1,\cdots,x_n]$, then how to get a bound of number of zeros of $g(f_1,\cdots,f_n)$ (in $\overline {\mathcal O_{K_v}}$) by the data of $A$, ${\bf f}(0)$ and $g$?

For example, if $f_i$s are polynomials, then the product of $\max_i\{\deg f_i\}\cdot \deg g$ gives a bound.

In general, by the theory of rigid geometry, I know we only need to calculate the order of $g({\bf f})$, which is the maximal index of the largest coefficients (using Weierstrass preparation theorem). But the order of analytic functions change weirdly while doing addition and subtraction (although it behaves well while doing multiplications). So how to calculate it?

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