Abstract
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.
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References
van den Dries L (1998) Tame topology and o-minimal structures. London Mathematical Society Lecture Note Series, 248. Cambridge University Press, Cambridge
van den Dries L, Miller C (1994) On the real exponential field with restricted analytic functions. Israel J Math 85(1–3): 19–56
Fratarcangeli S (2006) Rolle leaves and o-minimal structures. Doctoral Thesis
Gabrielov A (1996) Complements of subanalytic sets and existential formulas for analytic functions. Invent Math 125(1): 1–12
John F (1971) Partial differential equations. Applied Mathematical Sciences, vol 1. Springer, New York, viii+221 pp
Khovanskii AG (1980) A class of systems of transcendental equations. Dokl Akad Nauk SSSR 255: 762–765
Khovanskii AG (1991) Fewnomials. Translations of Mathematical Monographs, vol 88. American Mathematical Society, Providence, viii+139 pp
Macintyre A, Wilkie A (1996) On the decidability of the real exponential field. In: Kreiseliana, A. K. Peters (ed) Wellesley, MA, pp 441–467
Miller C (2001) Expansions of dense linear orders with the intermediate value property. J Symbol Logic 66(4): 1783–1790
Rolin J-P, Speissegger P, Wilkie AJ (2003) Quasianalytic Denjoy–Carleman classes and o-minimality. J Am Math Soc 16(4): 751–777
Sacks GE (1972) Saturated model theory. Mathematics Lecture Note Series. W. A. Benjamin, Inc., Reading, Mass. xii+335 pp
Servi T (2007) On the First Order Theory of Real Exponentiation. Doctoral thesis
Wilkie AJ (1996) Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function. J Am Math Soc 9: 1051–1094
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Servi, T. Noetherian varieties in definably complete structures. Log Anal 1, 187–204 (2008). https://doi.org/10.1007/s11813-008-0007-z
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DOI: https://doi.org/10.1007/s11813-008-0007-z