Artículo
Abstract:
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.
Registro:
| Documento: | Artículo |
| Título: | Effective Łojasiewicz inequalities in semialgebraic geometry |
| Autor: | Solernó, P. |
| Filiación: | Working Group Noaï Fitchas, IAM, Viamonte 1636. ler. piso., Buenos Aires, 1055, Argentina Fac. Ciencias Exactas y Naturales, Universidad de Buenos Aires, Argentina |
| Palabras clave: | Łojasiewicz inequalities; Complexity; Computer algebra; Real algebraic geometry |
| Año: | 1991 |
| Volumen: | 2 |
| Número: | 1 |
| Página de inicio: | 1 |
| Página de fin: | 14 |
| DOI: | http://dx.doi.org/10.1007/BF01810850 |
| Título revista: | Applicable Algebra in Engineering, Communication and Computing |
| Título revista abreviado: | AAECC |
| ISSN: | 09381279 |
| CODEN: | AAECE |
| Registro: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_09381279_v2_n1_p1_Solerno |
Referencias:
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- Ji, S., Kollár, J., Shiffman, B.: A global Łojasiewicz inequality for algebraic varieties. Trans. A.M.S. (to appear); Möller, H., Mora, F., Upper and lower bounds for the degree of Gröebner bases (1984) Proc. EUROSAM 84., Cambridge, England. Lecture Notes in Comp. Sci. Vol. 174, pp. 172-183. , Springer, Berlin, Heidelberg, New York
- Renegar, J.: On the computational complexity and geometry of the first order theory of the reals. Technical Report Vol. 856, Cornell University Ithaca (1989); Wüthrich, H., Ein Entscheidungsverfahren für die Theorie der reell-abgeschlossenen Körper (1976) Komplexität von Entscheidungsproblemen, Lecture Notes in Comp. Sci. Vol. 43, pp. 138-162. , Springer, Berlin, Heidelberg, New York
Citas:
---------- APA ----------
(1991) . Effective Łojasiewicz inequalities in semialgebraic geometry. Applicable Algebra in Engineering, Communication and Computing, 2(1), 1-14.http://dx.doi.org/10.1007/BF01810850
---------- CHICAGO ----------
Solernó, P. "Effective Łojasiewicz inequalities in semialgebraic geometry" . Applicable Algebra in Engineering, Communication and Computing 2, no. 1 (1991) : 1-14.http://dx.doi.org/10.1007/BF01810850
---------- MLA ----------
Solernó, P. "Effective Łojasiewicz inequalities in semialgebraic geometry" . Applicable Algebra in Engineering, Communication and Computing, vol. 2, no. 1, 1991, pp. 1-14.http://dx.doi.org/10.1007/BF01810850
---------- VANCOUVER ----------
Solernó, P. Effective Łojasiewicz inequalities in semialgebraic geometry. AAECC. 1991;2(1):1-14.http://dx.doi.org/10.1007/BF01810850
