Further notes on cell decomposition in closed ordered differential fields

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 (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 $\delta$-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 $CODF$.