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By Ehud Hrushovski

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Complement: Galois theory of ω-constructible sets. The definability theorem B1 presented above is somewhat weaker than the original presentation, as only constructible sets are allowed. We present here another version that permits also ωconstructible sets. We require stable embeddedness, but still very far from a global stability assumption. 1! Moreover, it is natural to work in an arbitrary universal domain for a universal theory T0 , not necessarily having a comprehensible completion. So we no longer assume QE (cf.

It 130 E. HRUSHOVSKI is stable so that every definable set (indeed every set) is stably embedded. C = C˜ is the equation Dx = 0. Q is the solution set to a linear differential equation. The torsor P in that case can be taken to be orbit of Aut(U/F, C) on the set of bases, an open subset of Qn ; the opposite group is the subgroup of elements Mn (C) preserving P and the action of Aut(U/F, C) on P , where Mn (C) is the group of n × n matrices with coefficients from C, acting on V n by matrix-vector multiplication.

Let D ⊂ U be Σ-constructible. Then Aut(U/D)-conjugacy is an ω-constructible equivalence relation. Proof. 1, we may assume D is stably embedded. In this case, two elements a, b are Aut(U/D)-conjugate iff for any constructible R and any tuple c of elements of D, one has R(a, c) ≡ R(b, c). This shows immediately that Aut(U/D)-conjugacy is ω-constructible. ω-constructible groups. Finally, we will mention without proof some background facts regarding definability of groups. By an ω-constructible group we mean a group whose universe is an ω-constructible set, and whose operations are constructible maps (maps whose graphs are constructible).

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[Article] Computing Galois group of a linear differential equation by Ehud Hrushovski

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