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...
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1991
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paper:paper_09381279_v2_n1_p1_Solerno2023-06-08T15:53:23Z Effective Łojasiewicz inequalities in semialgebraic geometry Solerno, Pablo Luis Łojasiewicz inequalities Complexity Computer algebra Real algebraic 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 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. Fil:Solernó, P. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. 1991 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_09381279_v2_n1_p1_Solerno http://hdl.handle.net/20.500.12110/paper_09381279_v2_n1_p1_Solerno |
| institution |
Universidad de Buenos Aires |
| institution_str |
I-28 |
| repository_str |
R-134 |
| collection |
Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) |
| topic |
Łojasiewicz inequalities Complexity Computer algebra Real algebraic geometry |
| spellingShingle |
Łojasiewicz inequalities Complexity Computer algebra Real algebraic geometry Solerno, Pablo Luis Effective Łojasiewicz inequalities in semialgebraic geometry |
| topic_facet |
Łojasiewicz inequalities Complexity Computer algebra Real algebraic geometry |
| description |
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. |
| author |
Solerno, Pablo Luis |
| author_facet |
Solerno, Pablo Luis |
| author_sort |
Solerno, Pablo Luis |
| title |
Effective Łojasiewicz inequalities in semialgebraic geometry |
| title_short |
Effective Łojasiewicz inequalities in semialgebraic geometry |
| title_full |
Effective Łojasiewicz inequalities in semialgebraic geometry |
| title_fullStr |
Effective Łojasiewicz inequalities in semialgebraic geometry |
| title_full_unstemmed |
Effective Łojasiewicz inequalities in semialgebraic geometry |
| title_sort |
effective łojasiewicz inequalities in semialgebraic geometry |
| publishDate |
1991 |
| url |
https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_09381279_v2_n1_p1_Solerno http://hdl.handle.net/20.500.12110/paper_09381279_v2_n1_p1_Solerno |
| work_keys_str_mv |
AT solernopabloluis effectivełojasiewiczinequalitiesinsemialgebraicgeometry |
| _version_ |
1768543954002771968 |