Further notes on cell decomposition in closed ordered differential fields

In [BMR05] the authors proved a theorem of cell decomposition for the theory CODF of closed ordered differential fields which generalize the usual Cell Decomposition Theorem for o-minimal structures. As a consequence of this result, a particularly well-behaving dimension function on definable sets in CODF was introduced. Here we carry on the study of this cell decomposition in CODF by proving three additional results. We first underline the relation between the δ -cells introduced in [BMR05] and the usual 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.