3 problems

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…

Bézivin's conjecture. If $Ly=0$ has a $\mathbb{Q}$-linearly independent basis of solutions in $\mathbb{Z}[[x]]$, then these solutions are algebraic over $\mathbb{Q}(x)$.

Grothendieck's $p$-curvature conjecture. If $L_py=0$ has a basis of $\mathbb{F}_p[[x^p]]$-linearly independent solutions in $\mathbb{F}_p[[x]]$ for almost all prime numbers $p$, th…