Noetherian varieties in definably complete structures
- 21 Downloads
We prove that the zero-set of a C ∞ function belonging to a noetherian differential ring M can be written as a finite union of C ∞ manifolds which are definable by functions from the same ring. These manifolds can be taken to be connected under the additional assumption that every zero-dimensional regular zero-set of functions in M consists of finitely many points. These results hold not only for C ∞ functions over the reals, but more generally for definable C ∞ functions in a definably complete expansion of an ordered field. The class of definably complete expansions of ordered fields, whose basic properties are discussed in this paper, expands the class of real closed fields and includes o-minimal expansions of ordered fields. Finally, we provide examples of noetherian differential rings of C ∞ functions over the reals, containing non-analytic functions.
KeywordsNoetherian varieties Definable completeness o-Minimality Quasi-analytic functions
Mathematics Subject Classification (2000)03C64 26E10 26E30
Unable to display preview. Download preview PDF.
- 1.van den Dries L (1998) Tame topology and o-minimal structures. London Mathematical Society Lecture Note Series, 248. Cambridge University Press, CambridgeGoogle Scholar
- 3.Fratarcangeli S (2006) Rolle leaves and o-minimal structures. Doctoral ThesisGoogle Scholar
- 5.John F (1971) Partial differential equations. Applied Mathematical Sciences, vol 1. Springer, New York, viii+221 ppGoogle Scholar
- 7.Khovanskii AG (1991) Fewnomials. Translations of Mathematical Monographs, vol 88. American Mathematical Society, Providence, viii+139 ppGoogle Scholar
- 8.Macintyre A, Wilkie A (1996) On the decidability of the real exponential field. In: Kreiseliana, A. K. Peters (ed) Wellesley, MA, pp 441–467Google Scholar
- 11.Sacks GE (1972) Saturated model theory. Mathematics Lecture Note Series. W. A. Benjamin, Inc., Reading, Mass. xii+335 ppGoogle Scholar
- 12.Servi T (2007) On the First Order Theory of Real Exponentiation. Doctoral thesisGoogle Scholar