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Some properties of the polynomially bounded o-minimal expansions of the real field and of some quasianalytic local rings. (English) Zbl 1472.03036

Let \(\mathcal E_n\) be the ring of germs of smooth functions in \(\mathbb R^n\) at the origin. The Borel map is a map from \(\mathcal E_n\) onto the ring of formal power series \(\mathbb R[[x_1,\ldots, x_n]]\). The image \(\widehat{f}\) of an \(f \in \mathcal E_n\) under the Borel map is defined as the infinite Taylor expansion of \(f\) at the origin. This paper discusses subrings \(\mathcal C\) of \(\mathcal E_n\) when the restrictions of the Borel map to \(\mathcal C\) are bijective. Its results are as follows:
Let \(\mathcal D_n\) be of the subring of \(\mathcal E_n\) consisting of the germs of real analytic functions definable in a polymonially bounded o-minimal expansion of the field of reals. The Weierstrass division and preparation theorems for \(\mathcal D_2\) hold true when the restriction of the Borel map to \(\mathcal D_1\) is bijective.
When a subring \(\mathcal C\) of \(\mathcal E_1\) satisfies the technical condition called the stability under monomial division and the restriction of the Borel map to it is bijective, the local ring \(\mathcal C\) with the \((x_1)\)-adic topology is complete.

MSC:

03C64 Model theory of ordered structures; o-minimality
14P99 Real algebraic and real-analytic geometry
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