Algebraicity, integrality, and prime-wise integrality of rational differential-equation solutions

Let $g\in \mathbb{Q}(z,y_0,\ldots,y_{n-1})$ be a rational function, let $t_0,\ldots,t_{n-1}\in\mathbb{Q}$ be initial values for which $g(0,t_0,\ldots,t_{n-1})$ is defined, and let

$$f(z)=\sum_{m\geq 0}a_mz^m\in\mathbb{Q}[[z]]$$

be the unique formal solution of $f^{(n)}(z)=g(z,f(z),\ldots,f^{(n-1)}(z))$ with $f^{(i)}(0)=t_i$. Algebraicity–integrality conjecture. The following are equivalent: $f(z)$ is algebraic over $\mathbb{Q}(z)$; there exists $N$ such that $a_i\in\mathbb{Z}[1/N]$ for all $i$; and there is a function $\omega(p):\operatorname{Primes}\to\mathbb{Z}$ with $\lim_{p\to\infty}\omega(p)/p=\infty$ such that $a_0,\ldots,a_{\omega(p)}\in\mathbb{Z}_{(p)}$ for every prime $p$. This is the paper's central conjecture, extending the Andr'e–Christol prediction beyond linear equations; it is presented as open, although several classical cases are proved.