Piotr Kowalski

Quantifier elimination for algebraic D-groups

We prove that if G is an algebraic D-group (in the sense of Buium over a differentially closed field (K,∂) of characteristic 0, then the first order structure consisting of G together with the algebraic D-subvarieties of G, G x G,..., has quantifier-elimination. In other words, the projection on G of a D-constructible subset of G n+1 is D-constructible. Among the consequences is that any finite-dimensional differential algebraic group is interpretable in an algebraically closed field.

Geometric axioms for existentially closed Hasse fields

Abstract We give geometric axioms for existentially closed Hasse fields. We prove a quantifier elimination result for existentially closed n -truncated Hasse fields and characterize them as reducts of existentially closed Hasse fields.

Integrating Hasse–Schmidt derivations

We study integrating (that is expanding to a Hasse–Schmidt derivation) derivations, and more generally truncated Hasse–Schmidt derivations, satisfying iterativity conditions given by formal group laws. Our results concern the cases of the additive and the multiplicative group laws. We generalize a theorem of Matsumura about integrating nilpotent derivations (such a generalization is implicit in work of Ziegler) and we also generalize a theorem of Tyc about integrating idempotent derivations.

A note on groups definable in difference fields

We prove that a group definable in a model of ACFA is virtually definably embeddable in an algebraic group. We give an improved proof of the same result for groups definable in differentially closed fields. We also extend to the difference field context results on the unipotence of definable groups on affine spaces.

Existentially closed fields with G-derivations

We prove that the theories of fields with Hasse-Schmidt derivations corresponding to actions of formal groups admit model companions. We also give geometric axiomatizations of these model companions.

Stable Groups and Algebraic Groups

A certain property of some type-definable subgroups of superstable groups with finite U-rank is closely related to the Mordell–Lang conjecture. This property is discussed in the context of algebraic groups.

AX–SCHANUEL CONDITION IN ARBITRARY CHARACTERISTIC

We prove a positive characteristic version of Axs theorem on the intersection of an algebraic subvariety and an analytic subgroup of an algebraic group. Our result is stated in a more general context of a formal map between an algebraic variety and an algebraic group. We derive transcendence results of Ax-Schanuel type.

A note on a theorem of Ax

Abstract We state and prove a generalization of Ax’s theorem on the transcendence degree of solutions of the differential equation of the exponential map. We also discuss a positive characteristic analogue of this theorem.

A note on extending actions of infinitesimal group schemes

We prove a non-integrability result concerning iterative derivations on projective line, where the iterative rule is given by a non-algebraic formal group.

Model theory of fields with virtually free group actions: MODEL THEORY OF FIELDS WITH VIRTUALLY FREE GROUP ACTIONS

For a group