Metric properties of semialgebraic mappings (Q309643)

Let \(X,Y\subset \mathbb{R}^{N}\) be closed semialgebraic sets, and suppose \( 0\in X\cap Y.\) The authors prove an effective estimation of the Łojasiewicz exponent for separation of these sets. That is there exist a neighbourhood \(U\subset \mathbb{R}^{N}\) of \(0\) and \(C>0\) such that \[ \mathrm{dist}(x,X)+\mathrm{dist}(x,Y)\geq C\mathrm{ dist}(x,X\cap Y)^{d(6d-3)^{N+r-1}}\text{ for }x\in U, \] where \(d\) and \(r\) are integers defined explicitly in terms of polynomial equations and inequalities describing \(X\) and \(Y.\) Precisely \(d\) is expressed in terms of degrees of these equations and inequalities, while \(r\) in terms of numbers of these inequalities. As a corollary they obtain estimations of the Łojasiewicz exponent for semialgebraic maps. Similar results in the same context are given for the Łojasiewicz exponents at infinity.