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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 $x(t)\in (\mathbb{Z}[t])^n$.
$\frac{dx(t)}{dt}=V(x(t)),x(0)=O\in \mathbb{R}^n$
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.

http://www.numdam.org/item/PMIHES_2001__93__161_0/

Since there may be counterexample, you can add freely aditional condtions.

I think this is a kind of non linear version of Grothendiek Katz conjecture.

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.

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 $x(t):\mathbb{R}\rightarrow \mathbb{R^n}$.
$\frac{dx(t)}{dt}=V(x(t)),x(0)=O\in \mathbb{R}^n$
If this equation has a polynomial solution in $\mathbb{Z}/p\mathbb{Z}[x_1,...,x_n]$ for every sufficiently large prime $p$,

Then does the original equation have polynomial solution in $\mathbb{Z}[x_1,...,x_n]$?

Here, a polynomial $f\in \mathbb{Z}/p\mathbb{Z}[x_1,...,x_n]$ 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.

http://www.numdam.org/item/PMIHES_2001__93__161_0/

Since there may be counterexample, you can add freely aditional condtions.
Other keyword
Grothendiek Katz conjecture

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 $x(t)\in (\mathbb{Z}[t])^n$.
$\frac{dx(t)}{dt}=V(x(t)),x(0)=O\in \mathbb{R}^n$
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.

http://www.numdam.org/item/PMIHES_2001__93__161_0/

Since there may be counterexample, you can add freely aditional condtions.

I think this is a kind of non linear version of Grothendiek Katz conjecture.

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 $x(t):\mathbb{R}\rightarrow \mathbb{R^n}$.
$\frac{dx(t)}{dt}=V(x(t)),x(0)=O\in \mathbb{R}^n$
If this equation has a polynomial solution in $\mathbb{Z}/p\mathbb{Z}[x_1,...,x_n]$ for every sufficiently large prime $p$,

Then does the original equation have polynomial solution in $\mathbb{Z}[x_1,...,x_n]$?

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.

http://www.numdam.org/item/PMIHES_2001__93__161_0/

Since there may be counterexample, you can add freely aditional condtions.
Other keyword
Grothendiek Katz conjecture

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 $x(t):\mathbb{R}\rightarrow \mathbb{R^n}$.
$\frac{dx(t)}{dt}=V(x(t)),x(0)=O\in \mathbb{R}^n$
If this equation has a polynomial solution in $\mathbb{Z}/p\mathbb{Z}[x_1,...,x_n]$ for every sufficiently large prime $p$,

Then does the original equation have polynomial solution in $\mathbb{Z}[x_1,...,x_n]$?

Here, a polynomial $f\in \mathbb{Z}/p\mathbb{Z}[x_1,...,x_n]$ 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.

http://www.numdam.org/item/PMIHES_2001__93__161_0/

Since there may be counterexample, you can add freely aditional condtions.
Other keyword
Grothendiek Katz conjecture