Are there any good reference to tackle the problem below? Or, are there any know result?
Problem
Let $f_1...f_n\in \mathbb{Z}[x_1,..,x_n]$ and $V:\mathbb{R}^n\rightarrow \mathbb{R}^n$ be a vector field on $\mathbb{R}^n$ defined by each $f_i$.
Let's consider a initial value problem for $x(t)\in (\mathbb{Z}[t])^n$.
$$
\begin{cases}
\dfrac{dx(t)}{dt}=V(x(t)),\\
x(0)=\mathbf 0\in \mathbb{R}^n
\end{cases}
$$
Quesstion. If this equation has a polynomial solution in $(\mathbb{Z}/p\mathbb{Z}[t])^n$ for every sufficiently large prime $p$, then does the original equation have polynomial solution in $(\mathbb{Z}[t])^n$?
Here, $F\in (\mathbb{Z}/p\mathbb{Z}[t])^n$ is a solution of this equation, if $F$ is a formal solution of a differential equation with $\tilde{V}: (\mathbb{Z}/p\mathbb{Z})^n\rightarrow (\mathbb{Z}/p\mathbb{Z})^n$ which is a reduction of $V$ mod $p$
Moreover what if $V$ is $p$-closed for each sufficiently large $p$ i.e. $V^p=0$ as derivation on $\mathbb{Z}/p\mathbb{Z}[x_1,...,x_n]$.
Bost's paper about algebraicity of foliaiton seems related to this question but I don't know whether I can I apply techniques used there.
Full reference:
Jean-Benoît Bost, "Algebraic leaves of algebraic foliations over number fields", Publications Mathématiques de l'IHÉS, Tome 93 (2001), pp. 161-221. MR1863738, Zbl 1034.14010.
Since there may be counterexamples, you can freely add additional conditions.
I think this is a kind of non linear version of Grothendieck-Katz conjecture.
