The model theory of m‐ordered differential fields

Abstract

Key words m-ordered differential fields, model companion, geometric axiomatization. MSC (2000) 03C10, 12J15In his Ph. D. thesis [7], L. van den Dries studied the model theory of fields (more precisely domains) with finitely many orderings and valuations where all open sets according to the topology defined by an order or a valuation is globally dense according with all other orderings and valuations. Van den Dries proved that the theory of these fields is companionable and that the theory of the companion is decidable (see also [8]).In this paper we study the case where the fields are expanded with finitely many orderings and an independent derivation. We show that the theory of these fields still admits a model companion in the language L D <,m = {+, −, ·, D, <1, . . . , <m, 1, 0}. We denote this model companion by CODFm and give a geometric axiomatization of this theory which uses basic notions of algebraic geometry and some generalized open subsets which appear naturally in this context. This axiomatization allows to recover (just by putting m = 1) the one given in [4] for the theory CODF of closed ordered differential fields. Most of the technics we use here are already present in [2] and [4].Finally, we prove that it is possible to describe the completions of CODFm and to obtain quantifier elimination in a slightly enriched (infinite) language. This generalizes van den Dries' results in the "derivation free" case.In order to make this paper as self-contained as possible, we briefly recall the results of [7].Let m ∈ N, an m-ordered field M is a field equipped with m orderings < 1 , . . . , < m . Any such field is an L <,m -structure, where L <,m := {+, ·, −, < 1 , . . . , < m , 0, 1}. We denote by OF m the L <,m -theory of m-ordered fields. Fact 1.1 [7, Chapter II, Theorem (1.2)] OF m has a model companion COF m whose models M satisfy: (1) OF m . (2) The orderings on M are pairwise independent 1) . (3) For any irreducible f (X, Y ) ∈ M [X, Y ] and any a ∈ M such that f (a, Y ) changes sign on M w. r. t. to each ordering, there exists (c, d) ∈ M 2 with f (c, d) = 0. The independence of the orderings leads to some interesting consequences. Let M be a model of COF m , a subset U of M k is m-open if U = U 1 ∩ · · · ∩ U m , where, for each i ∈ {1, . . . , m}, U i is a < i -open 2) subset of M

Topological differential fields

“…In Section 11, we will give the application to pseudo-finite fields. It will also allow us to obtain new results on differential fields endowed with several valuations and to give an alternative axiomatization of the model-companion in the case of differential e-fold ordered fields (see [32]). …”

“…The van den Dries fields form an inductive class of topological L e,< -fields. The only non-trivial point to check is the Hypothesis (I) (see Corollary 2.14 in [32]). One shows that given any e-fold ordered field K and the field of Laurent series L := K ((t)), one can extend the e-orderings to L. Then given an element f [X] ∈ L[X ] and a ∈ L such that f (a) ∼ 0 and f 2 (a) ∼ 0, we can find a zero b with a ∼ b in the following way.…”

“…In [32] Rivière considered the maximal PRC e fields expanded with a derivation D with no interaction with the interval topologies. First he showed that the theory of these fields still admits a model-companion in the language L e,< ∪ {D} and gave a geometric axiomatization of this model-companion.…”

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].…”

Generic derivations on o-minimal structures

“…People have extended DCF 0 in another direction by considering fields which are not algebraically closed: Singer, and later others [Sin78, Poi11, BCKP19, BMR09, Riv09] studied real closed fields with one generic derivation, and [Riv06b] extended to m commuting derivations (see also [FK20] for a different approach); [GP10, GP12, GR06, CKP23] studied more general topological fields with one generic derivation. In [Riv06a] the author studied fields with m independent orderings and one generic derivation and in [FK20] they studied o-minimal structures with several commuting generic "compatible" derivations. In her PhD thesis, Borrata [Bor21] studied ordered valued fields and "tame" pairs of real closed fields endowed with one generic derivation.…”