Abstract
Let S be a semialgebraic set given by a boolean combination of polynomial inequalities. We present an algorithmical method which solves in single exponential sequential time and polynomial parallel time, the following problems:
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computation of the dimension of S.
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computation of the number of semialgebraically connected components of S and construction of paths in S connecting points in the same component.
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computation of the distance of S to another semialgebraic set and finding points realizing the distance if they exist.
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computation of the “optical resolution” of S if S is closed (the pelotita and the bolón).
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computation of integer Morse directions of S if S is a regular algebraic hypersurface.
The mentioned time bounds apply also to polynomial inequalities solving. As an application of our method we state an efficient Łojasiewicz inequality and an efficient finiteness theorem.
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Ben-Or M., Kozen D., Reif J.: The complexity of elementary algebra and geometry. J. of Comp. and Syst. Sci. 32 (1986) 251–264
Bochnak J., Coste M., Roy M-F.: Géométrie algébrique réelle. Springer-Verlag (1987)
Brownawell W.: Bounds for the degrees in the Nullstellensatz. Ann. Math. Vol. 126 No 3 (1987) 287–290
Caniglia L., Galligo A., Heintz J.: Some new effectivity bounds in computational geometry. Proc. AAECC-6 LN Comp. Sci. 357 (1988) 131–151
Caniglia L., Galligo A., Heintz J.: Borne simple exponentielle pour les degrés dans le théorème des zéros sur un corps de caractéristique quelconque. CR Acad. Sci. Paris, t.307 (1988) 255–258
Canny J.: The complexity of motion planning. MIT Thesis 1986, MIT Press (1988)
Canny J.: A new algebraic method for robot motion planning and real algebraic geometry. Proc. 28th FOCS (1987) 39–48
Canny J., Grigor'ev D.Yu., Vorobjov N.N. (Jr.): Finding connected components of a semialgebraic set in subexponential time. Manuscript Steklov Math. Inst. Leningrad LOMI (1990)
Coste M., Roy M-F.: Thom's Lemma, the coding of real algebraic numbers and the topology of semialgebraic sets. J. of Symb. Comp. 5 (1988) 121–129
Dickenstein A., Fitchas N., Giusti M., Sessa C.: The membership problem for unmixed polynomial ideals is solvable in single exponential time. To appear in Proc. AAECC 7, Toulouse (1989)
Fitchas N., Galligo A., Morgenstern J.: Algorithmes en séquentiel et en parallèle pour l'élimination des quantificateurs en géométrie élémentaire. Séminaire Structures Algébriques Ordonnées, Sélection d'exposés 1984–1987 Vol I. Publ. Univ. Paris VII, No. 32, (1990) 29–35
Fitchas N., Galligo A., Morgenstern J.: Precise sequential and parallel complexity bounds for the quantifier elimination of algebraically closed fields. Journal of Pure and Applied Algebra 67 (1990) 1–14
Fitchas N., Galligo A.: Nullstellensatz effectif et conjecture de Serre (Théorème de Quillen-Suslin) pour le Calcul Formel. Sém. Structures Alg. Ord. Univ. Paris VII; final version to appear in Math. Nachrichten
Fulton W.: Intersection Theory. Springer Verlag (1984)
von zur Gathen J.: Parallel arithmetic computations: a survey. Proc. 13th Conf. MFCS (1986)
González L., Lombardi H., Recio T., Roy M-F.: Sous-résultants et spécialisations de la suite de Sturm. To appear in RAIRO, Inf. Théorique.
Grigor'ev D. Yu.: Complexity of deciding Tarski algebra. J. Symb. Comp. 5 (1988) 65–108
Grigor'ev D.Yu., Heintz J., Roy M-F., Solernó P., Vorobjov N.N. (Jr.): Comptage des composantes connexes d'un ensemble semi-algébrique en temps simplement exponentiel. To appear in C.R. Acad. Sci. Paris
Grigor'ev D.Yu., Vorobjov N.N. (Jr.): Solving systems of polynomial inequalities in subexponential time. J. Symb. Comp. 5 (1988) 37–64
Grigor'ev D.Yu., Vorobjov N.N. (Jr.): Counting connected components of a semialgebraic set in subexponential time. Manuscript Steklov Math. Inst. Leningrad LOMI (1990)
Guillemin V., Pollack A.: Differential Topology. Prentice-Hall (1974)
Heintz J.: Definability and fast quantifier elimination over algebraically closed fields. Theor. Comp. Sci. 24 (1983) 239–277
Heintz J., Roy M-F., Solernó P.: Complexity of semialgebraic sets. Manuscript IAM (1988)
Heintz J., Roy M-F., Solernó P.: On the complexity of semialgebraic sets (ext. abst.). Proc. IFIP'89, 293–298
Heintz J., Roy M-F., Solernó P.: Complexité du principe de Tarski-Seidenberg. C.R. Acad. Sci. Paris t.309 (1989) 825–830
Heintz J., Roy M-F., Solernó P.: Sur la complexité du principe de Tarski-Seidenberg. Bull. de la Soc. Math. de France 118 (1990) 101–126
Heintz J., Roy M-F., Solernó P.: Single exponential path finding in semialgebraic sets. Preprint (1990)
Heintz J., Roy M-F., Solernó P.: Construction de chemins dans un ensemble semialgébrique. Manuscript (1990)
Ji S., Shiffman B.: A global Łojasiewicz inequality for complete intersections in Cn. Preprint (1989)
Kollár J.: Sharp effective Nullstellensatz. J. AMS 1 (1988) 963–975
Kollár J.: A Łojasiewicz-type Inequality for Algebraic Varieties. Preprint (1989)
Milnor J.: Morse Theory. Princeton Univ. Press (1963)
Renegar J.: On the computational complexity and geometry of the first order theory of the reals I, II, III Technical Reports 853, 854, 856 (1989)
Solernó P.: Complejidad de conjuntos semialgebraicos. Thesis Univ. de Buenos Aires (1989)
Solernó P.: Construction de fonctions de Morse pour une hypersurface régulière en temps admissible. Manuscript IAM (1989)
Solernó P.: Effective Lojasiewicz inequalities. To appear in AAECC Springer-Verlag (1990)
Teissier B.: Résultats récents d'algèbre commutative effective. Séminaire Bourbaki 718 (1989)
Trotman D.: On Canny's roadmap algorithm: orienteering in semialgebraic sets. Manuscript Univ. Aix-Marseille (1989)
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Heintz, J., Krick, T., Roy, MF., Solernó, P. (1991). Geometric problems solvable in single exponential time. In: Sakata, S. (eds) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. AAECC 1990. Lecture Notes in Computer Science, vol 508. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-54195-0_35
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DOI: https://doi.org/10.1007/3-540-54195-0_35
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