Effective Łojasiewicz inequalities in semialgebraic geometry

Abstract

The main result of this paper can be stated as follows: letV ⊂ ℝⁿ be a compact semialgebraic set given by a boolean combination of inequalities involving only polynomials whose number and degrees are bounded by someD > 1. LetF, G∈∝[X₁,⋯, X_n] be polynomials with degF, degG ≦ D inducing onV continuous semialgebraic functionsf, g:V→R. Assume that the zeros off are contained in the zeros ofg. Then the following effective Łojasiewicz inequality is true: there exists an universal constantc ₁∈ℕ and a positive constantc ₂∈∝ (depending onV, f,g) such that$|g(x)|^{D^{c_1 .n} } \leqq c_2 \cdot |f(x)|$ for allx∈V. This result is generalized to arbitrary given compact semialgebraic setsV and arbitrary continuous functionsf,g:V → ∝. An effective global Łojasiewicz inequality on the minimal distance of solutions of polynomial inequalities systems and an effective Finiteness Theorem (with admissible complexity bounds) for open and closed semialgebraic sets are derived.