Model completion of Lie differential fields

Model theory of partial differential fields: From commuting to noncommuting derivations

“…In [7], Yaffe defines the universal theory LDF 0 of Lie differential fields. To do this he fixes a field F of characteristic zero, a finite-dimensional F-vector space L with a Lie multiplication making it a Lie algebra over a subfield of F, and a vector space homomorphism φ F : L → Der(F), the Lie algebra of derivations on F, preserving the Lie multiplication.…”

“…Independently, Yaffe [7] gave axioms for the model completion of the universal theory of a more general class of fields: LDF 0 , the universal theory of fields of characteristic zero together with a finite-dimensional Lie algebra acting as derivations. The goal of this short note is to show that starting from McGrail's results one can quickly write down axioms for Yaffe's model completion, that is, one can reduce the noncommutative case to the commutative case.…”

Differential arcs and regular types in differential fields

“…In the case of DCF 0,1 there is only one infinite rank regular type (up to nonorthogonality), namely the generic type of the ambient differentially closed field, for which Zilber's conjecture clearly holds. However in the case of PDE's , that is in the case of DCF 0,n for n > 1 (which is superstable [9,16]), there will be many infinite rank regular types. It is worth saying at this point what regularity amounts to in DCF 0,n : The generic type of an irreducible differential variety X will be regular if whenever (Y a : a ∈ Z) is a differential algebraic family of irreducible proper differential subvarieties of X, whose union is Kolchin dense in X, then any generic member Y a of the family is orthogonal to X.…”

Differential algebra for derivations with nontrivial commutation rules

“…Indeterminates can be introduced one by one. A first extension consists in considering (finite sets) the derivations on F that satisfy nontrivial commutation rules [22,40]. The considered derivations generate a subspace of the F-vector space of derivations on F. An algebraic version of Frobenius theorem [11,Proposition 39;22, Section 0.5, Proposition 6] shows that you can always choose a commuting basis of derivations for that vector subspace provided it is closed under the commutator.…”