Definable groups in topological fields with a generic derivation

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

logicnumber theory

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Online logic seminar

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