Markov’s inequality in the o-minimal structure of convergent generalized power series. (English) Zbl 1271.32014
The author introduces (via some convergent generalized power series) an o-minimal structure \(\mathcal S\) which generates a new large class of sets with the following UPC property.
Assume that \(E\subset\mathbb R^k\) is a bounded fat and definable set in \(\mathcal S\). Then there exist \(v, M>0\), and \(d\in\mathbb N\) such that for each \(z\in\overline E\) there exists a polynomial map \(h_z:\mathbb R\longrightarrow\mathbb R^k\) of degree \(\leq d\) such that \(h_z(0)=z\) and dist\((h_z(t), \mathbb R^k\setminus E)\geq Mt^v\) for all \(t\in[0,1]\).
Assume that \(E\subset\mathbb R^k\) is a bounded fat and definable set in \(\mathcal S\). Then there exist \(v, M>0\), and \(d\in\mathbb N\) such that for each \(z\in\overline E\) there exists a polynomial map \(h_z:\mathbb R\longrightarrow\mathbb R^k\) of degree \(\leq d\) such that \(h_z(0)=z\) and dist\((h_z(t), \mathbb R^k\setminus E)\geq Mt^v\) for all \(t\in[0,1]\).
Reviewer: Marek Jarnicki (Kraków)
MSC:
| 32E30 | Holomorphic, polynomial and rational approximation, and interpolation in several complex variables; Runge pairs |
| 32B20 | Semi-analytic sets, subanalytic sets, and generalizations |
| 03C64 | Model theory of ordered structures; o-minimality |
