An o-minimal structure which does not admit \(C^\infty\) cellular decomposition.

*(English)*Zbl 1193.03065The aim of this well-written paper is to construct an o-minimal expansion of the real field which does not admit \(C^\infty\)-cell decomposition, thus settling an old open question in the subject.

It is not too hard to construct functions on the reals which are not \(C^\infty\) at the origin but are \(C^k\) for any \(k\), and further, are piecewise polynomial away from \(0\). The problem is to show that some such function is o-minimal.

The authors follow the setting of [J.-P. Rolin, P. Speissegger and A. J. Wilkie, J. Am. Math. Soc. 16, No. 4, 751–777 (2003; Zbl 1095.26018)] and so have to show that large algebras of functions generated by the initial function, under some basic operations, are quasianalytic. The main work, and difficulty, in the paper is to show that if the Taylor expansion of the initial function at the origin is sufficiently generic then the desired quasianalyticity can be achieved.

The authors then adapt the model completeness and o-minimality proofs from the paper cited above to work with suitable non-\(C^{\infty}\) functions, and this completes the proof.

It is not too hard to construct functions on the reals which are not \(C^\infty\) at the origin but are \(C^k\) for any \(k\), and further, are piecewise polynomial away from \(0\). The problem is to show that some such function is o-minimal.

The authors follow the setting of [J.-P. Rolin, P. Speissegger and A. J. Wilkie, J. Am. Math. Soc. 16, No. 4, 751–777 (2003; Zbl 1095.26018)] and so have to show that large algebras of functions generated by the initial function, under some basic operations, are quasianalytic. The main work, and difficulty, in the paper is to show that if the Taylor expansion of the initial function at the origin is sufficiently generic then the desired quasianalyticity can be achieved.

The authors then adapt the model completeness and o-minimality proofs from the paper cited above to work with suitable non-\(C^{\infty}\) functions, and this completes the proof.

Reviewer: Gareth Jones (Manchester)

##### MSC:

03C64 | Model theory of ordered structures; o-minimality |

26A27 | Nondifferentiability (nondifferentiable functions, points of nondifferentiability), discontinuous derivatives |

57R45 | Singularities of differentiable mappings in differential topology |

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