The theory of closed ordered differential fields with m commuting derivations

Mini-Workshop: Surreal Numbers, Surreal Analysis, Hahn Fields and Derivations

“…(11) Existence and uniqueness of Picard-Vessiot extensions of CODFs have been shown in [31] and generalised in [66]. There is also a model completion with quantifier elimination of ordered fields equipped with several commuting derivatives, see [107] and [92] for an explicit axiomatisation. Again, these structures have the NIP, but apart from that much less is known compared to the ordinary case.…”

Model Theory of Fields With Operators – a Survey

“…This was done by Singer ([94]) for ordered differential fields (thus obtaining a notion of closed ordered differential field), and by Tressl for the class of differential fields which are large, and whose theory in the language of rings is model-complete 1 . More work on closed ordered differential fields was done by Brihaye, Michaux, Point, and Rivière ([64], [78], [80], [81], [82]). The "uniform" axiomatisation of Tressl was generalised by Guzy in [34].…”

A nullstellensatz and a positivstellensatz for ordered differential fields

“…, θ k (ū)) for someū ∈ K n . We will work in the theory m-CODF of closed ordered ∆-fields studied and axiomatised by C. Rivière in [10] and M. Tressl in [16]. They showed that m-CODF is the model completion of the universal theory of ordered fields endowed with m commuting derivations and so eliminates quantifiers.…”

“…In the last section, using T. McGrail and C. Rivière's results (see [7] and [10]) on partial differential fields and ordered partial differential fields, we explain how our results can be generalised to that context.…”

“…These fields are called ordered partial differential fields and their model completion is called m-CODF (where m is the number of derivations put on the field). M. Tressl and C. Rivière studied this theory and also gave an axiomatisation similar to the case of one derivation (see [16] and [10]). …”