Definable groups in topological fields with a generic derivation

Françoise Point (Université de Mons-Hainaut)

07-Oct-2021, 18:00-19:00 (3 years ago)

Abstract: We study a class of tame L\mathcal L-theories TT of topological fields and their extensions by a generic derivation δ\delta. The topological fields under consideration include henselian valued fields of characteristic 0 and real closed fields. We axiomatize the class of the existentially closed Lδ\mathcal L_\delta-expansions. We show that TδT_\delta^* has L\mathcal L-open core (i.e., every Lδ\mathcal L_\delta-definable open set is L\mathcal L-definable) and derive both a cell decomposition theorem and a transfer result of elimination of imaginaries. Other tame properties of TT such as relative elimination of field sort quantifiers, NIP and distality also transfer to TδT_\delta^*. \par Then letting K\mathcal K be a model of TδT_\delta^* and M\mathcal M a K+\vert K\vert^+-saturated elementary extension of K\mathcal K, we first associate with an Lδ(K)\mathcal L_\delta(K)-definable group GG in M\mathcal M, a pro-L\mathcal L-definable set GG^{**}_{\infty} in which the differential prolongations GG^{\nabla_\infty} of elements of GG are dense, using the L\mathcal L-open core property of TδT_\delta^*. Following the same ideas as in the group configuration theorem in o-minimal structures as developed by K. Peterzil, we construct a type L\mathcal L-definable topological group HGH_\infty\subset G^{**}_{\infty}, acting on a KK-infinitesimal neighbourhood of a generic element of GG^{**}_\infty in a faithful, continuous and transitive way. Further HGH_\infty\cap G^{\nabla_\infty} is dense in HH_\infty and the action of HGH_\infty\cap G^{\nabla_\infty} coincides with the one induced by the initial Lδ\mathcal L_\delta-group action. \par The first part of this work is joint with Pablo Cubid\`es Kovacsics.

logicnumber theory

Audience: researchers in the topic


Online logic seminar

Series comments: Description: Seminar on all areas of logic

Organizer: Wesley Calvert*
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