Further notes on cell decomposition in closed ordered differential fields

In [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 ( C O D F ) 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 C O D F was introduced. Here we continue the study of this cell decomposition in C O D F 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 C O D F .