an:05563894
Zbl 1166.03017
Rivi??re, C??dric
Further notes on cell decomposition in closed ordered differential fields
EN
Ann. Pure Appl. Logic 159, No. 1-2, 100-110 (2009).
00250042
2009
j
03C64 12H05 12J15 12L12
ordered differential fields; cell decomposition; dimensional polynomial; o-minimal structures
Summary: In [\textit{T. Brihaye, C. Michaux} and \textit{C. Rivi??re}, ``Cell decomposition and dimension function in the theory of closed ordered differential fields'', Ann. Pure Appl. Logic 159, No.~1--2, 111--128 (2009; Zbl 1166.03015)], 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.
Zbl 1166.03015