Further notes on cell decomposition in closed ordered differential fields

Abstract

a b s t r a c tIn [T. Brihaye, C. Michaux, C. Rivière, Cell decomposition and dimension function in the theory of closed ordered differential fields, Ann. Pure Appl. Logic (in press).] the authors proved a cell decomposition theorem for the theory of closed ordered differential fields (CODF ) which generalizes the usual Cell Decomposition Theorem for o-minimal structures. As a consequence of this result, a well-behaving dimension function on definable sets in CODF was introduced. Here we continue the study of this cell decomposition in CODF by proving three additional results. We first discuss the relation between the δ-cells introduced in the above-mentioned reference and the notion of Kolchin polynomial (or dimensional polynomial) in differential algebra. We then prove two generalizations of classical decomposition theorems in o-minimal structures. More exactly we give a theorem of decomposition into definably d-connected components (d-connectedness is a weak differential generalization of usual connectedness w.r.t. the order topology) and a differential cell decomposition theorem for a particular class of definable functions in CODF . OutlineThis paper is in direct filiation with paper [1]. Even though we recall in Section 1 some of the developments of the previous paper, it is certainly helpful to have a look at it before reading this one. In the sequel, we will denote by L the language {+, −, * , <, 0, 1} of ordered rings and by L the language {+, −, * , , <, 0, 1} of ordered differential rings.The first section of this paper contains a brief summary of the work presented in [1]. In the latter, the authors study a differential analogue of o-minimality in the theory CODF of closed ordered differential fields. In particular we recall the statement of the differential cell decomposition theorem for definable sets in CODF (Theorem 1.6). Section 2 was motivated by a question of T. Scanlon and contains the developments required to link the notion of δ-cell introduced in [1] with the Kolchin polynomial defined in partial 1 differential algebra [2, Theorem 6,p.115]. In the particular case of a differential field M equipped with a single derivation, the Kolchin polynomial describes, for any tupleā in an extension of M, the asymptotic behavior of the algebraic transcendence degree of the field M(ā,ā , . . . ,ā (n) ) over M (when n tends to ∞). Furthermore W.-Y. Pong proved in [4] that this polynomial has a very simple form dX + b where d is the differential transcendence degree ofā over M and b is a positive integer. Our aim here is to explain how some further investigations concerning δ-cells allow recovering the integers d and b (and then the Kolchin polynomial) in case M is a model of CODF . For this we first define a notion of K -type for a particular class of δ-cells called engaged δ-cells (Definitions 2.4-2.6). In fact the K -type provides a rank on δ-cells which is more precise than the δ-dimension and allows associating a K -rank with any tupleā in a differential extension of M (Definition 2.8). We finally pro...

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

“…See [105]. (4) There is a cell decomposition theorem and a notion of geometric differential dimension which agrees with transcendence degree on types, see [18], [93]. (5) CODF has an o-minimal open core, see [89] who deduces this from (4).…”

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].…”