Difference Galois inverse problem answered in the affirmative: every linear algebraic group with cyclic component quotient becomes a Picard–Vessiot group over the shift
Synopsis
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.
Interpretation
The range of Galois groups realizable by shift equations σY=A(x)Y is completed: any linear algebraic group G whose component quotient G/G° is cyclic occurs as the Picard–Vessiot group of such a system. Van der Put and Singer had shown that a Picard–Vessiot group necessarily has cyclic G/G°, and stated the converse as a conjecture; known positive realizations covered only connected groups, cyclic extensions of tori and similar classes. The proof is explicit: finitely many generating points of G° are interpolated into a rational curve B(z), the matrix is set to A(x)=hB(x/t) with h semisimple generating G/G°, and the coefficient group TA is verified to equal G.
The construction makes the coefficient matrix free of an x⁻¹ term at infinity and gives its finite singularities an arithmetic condition, which rules out rational changes of variables reducing the system to a proper subgroup. Counterexamples show the conclusion fails without these conditions: a unipotent limit gives a trivial Picard–Vessiot group while the coefficient group is the additive group, and counting only poles of A rather than of A⁻¹ gives a wrong singularity set. Under A(x)=h+O(x⁻²) with h semisimple and 1 outside the rational span of the singularity set, every rational difference-invariant line in every representation is the scalar extension of a constant line; combined with the coefficient-group criterion this forces the Galois group to be G.
Perspective
The result concerns C(x) over an algebraically closed field C of characteristic zero with the shift σ(x)=x+1, and realizes closed subgroups G⊆GLn(C) with G/G° cyclic; the analogous inverse problem for the q-shift σq(f(x))=f(qx) is outside the discussion.
The proof relies on additional conditions: vanishing of the first-order term at infinity, semisimplicity of the limit h, and a singularity set whose rational span excludes 1; counterexamples in the text show the conclusion fails once semisimplicity or the singularity condition is dropped.