Abstract

In [T. Brihaye, C. Michaux, C. Rivière, Cell decomposition and dimension function in the theory of closed ordered differential fields, Ann. Pure Appl. Logic .] the authors proved a cell decomposition theorem for the theory of closed ordered differential fields 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 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 and a differential cell decomposition theorem for a particular class of definable functions in CODF