Difference Galois inverse problem answered in the affirmative: every linear algebraic group with cyclic component quotient becomes a Picard–Vessiot group over the shift
For the rational function field C(x) under the shift σ(x)=x+1, it was known that a Picard–Vessiot group G must have G/G° cyclic, but which groups are actually realizable remained open. It is now shown that every closed subgroup G⊆GLn(C) with cyclic G/G° occurs, with coefficient matrix of the form A(x)=h+O(x⁻²) at infinity for semisimple h, answering the conjecture of van der Put and Singer.