8 problems

Let $f(x)$ be a globally bounded D-finite series, meaning a D-finite series with positive radius of convergence that can be reduced modulo almost all primes. A multivariate power s…

Automaticity conjecture. There is a polynomial $P\in\mathbb{F}_q(t)[x]$ such that

Let $F(\mathbf{x})$ be a multivariate algebraic power series with integer coefficients, and let $\operatorname{Diag}(F)$ denote its diagonal. For each prime number $p$, reduce the…

Let $f(x)$ be a $G$-function, and suppose that its reduction modulo $p$ is defined for infinitely many primes $p$. Adamczewski–Delaygue conjecture. For almost all such primes $p$,…

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…

Let $R$ be a ring for which a “smallest” Henselian extension can be defined, and let $\widehat{R}$ denote its completion. The algebraic closure of $R$ in $\widehat{R}$ consists of…