Topological fields with a generic derivation

2019/12/17 by Kovacsics, Pablo Cubides, Point, Françoise · 1 citation
#03C60 #12H05 #12J25 #Commutative Algebra (math.AC) #FOS: Mathematics #Logic (math.LO) #Primary 12L12 #Secondary 13N15

paper · doi:10.48550/arxiv.1912.07912

Abstract

We study a class of tame L-theories T of topological fields and their L_δ-extension T_δ^* by a generic derivation δ. The topological fields under consideration include henselian valued fields of characteristic 0 and real closed fields. We show that the associated expansion by a generic derivation has L-open core (i.e., every L_δ-definable open set is 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_δ^*. As an application, we derive consequences for the corresponding theories of dense pairs. In particular, we show that the theory of pairs of real closed fields (resp. of p-adically closed fields and real closed valued fields) admits a distal expansion. This gives a partial answer to a question of P. Simon.

Cited by

• Generic derivations on algebraically bounded structures II Model theoretical properties

Related

• Definable groups in topological fields with a generic derivation
• On definable groups in dp-minimal topological fields equipped with a generic derivation
• Definable groups in topological differential fields
• The algebra and model theory of tame valued fields
• On Lambda functions in henselian and separably tame valued fields