Effective Łojasiewicz inequalities in semialgebraic geometry
The main result of this paper can be stated as follows: let V ⊂ ℝn be a compact semialgebraic set given by a boolean combination of inequalities involving only polynomials whose number and degrees are bounded by some D > 1. Let F, G∈∝[X1,⋯, Xn] be polynomials with deg F, deg G ≦ D inducing on...
Guardado en:
| Autor principal: | |
|---|---|
| Formato: | JOUR |
| Materias: | |
| Acceso en línea: | http://hdl.handle.net/20.500.12110/paper_09381279_v2_n1_p1_Solerno |
| Aporte de: |
| Sumario: | The main result of this paper can be stated as follows: let V ⊂ ℝn be a compact semialgebraic set given by a boolean combination of inequalities involving only polynomials whose number and degrees are bounded by some D > 1. Let F, G∈∝[X1,⋯, Xn] be polynomials with deg F, deg G ≦ D inducing on V continuous semialgebraic functions f, g:V→R. Assume that the zeros of f are contained in the zeros of g. Then the following effective Łojasiewicz inequality is true: there exists an universal constant c1∈ℕ and a positive constant c2∈∝ (depending on V, f,g) such that {Mathematical expression} for all x∈V. This result is generalized to arbitrary given compact semialgebraic sets V and arbitrary continuous functions f,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. © 1991 Springer-Verlag. |
|---|