3 problems

Conjectural description. The difference Galois group is the smallest algebraic subgroup $G$ of ${\rm GL}(V)$ such that $G$ contains all $G_j$ and the Lie algebra of $G$ contains al…

van der Put–Singer inverse problem conjecture. If $G/G^\circ$ is cyclic, then $G$ is the Galois group of a linear difference equation

Let $F={\mathbb{C}}(z)$ and let $\phi:F\to F$ be given by $\phi(z)=z+1$. For a linear algebraic group ${\mathcal{G}}$ over ${\mathbb{C}}$, a difference Galois group means the Galoi…