Abstract
This paper offers a short survey of two topics. In the first section we describe
a new bound on the Betti numbers of semi-algebraic sets whose proof rely on
the Thom-Milnor bound for algebraic sets and the Mayer-Vietoris long exact
sequence. The second section describes the best complexity results and practical
implementations currently available for finding real solutions of systems of
polynomial equations and inequalities. In both sections, the consideration of
critical points will play a key role.
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References
N. Alon The number of polytopes, configurations and real matroids. Mathematika, 33, 62–71 (1986).
P. Aubry, F. Rouillier, M. Safey El Din Real solving for real algebraic varieties defined by several equations. (2000), submitted.
S. Basu, R. Pollack, M.-F. Roy, On the number of cells defined by a family of polynomials on a variety. Mathematika, 43, 120–126 (1996).748 M.-F. Roy
S.Basu, R. Pollack, M.-F. Roy, On the Combinatorial and Algebraic Complexity of Quantifier Elimination. Journal of the ACM, 43, 1002–1045 (1996).
[5]S. Basu, R. Pollack, M.-F. Roy, On Bounding the Betti Numbers of the Connected Components Defined by a Set of Polynomials and of Closed Semialgebraic Sets. In preparation.
[6]S. Basu, R. Pollack, M.-F. Roy, Algorithms in real algebraic geometry. Book in preparation.
[7]J. Bochnak, M. Coste, M.-F. Roy, Real algebraic geometry (second edition, Springer-Verlag
J. Canny, The Complexity of Robot Motion Planning. MIT Press (1987).
J. Canny, Some Algebraic and Geometric Computations in PSPACE. Proc.Twentieth ACM Symp. on Theory of Computing, 460–467 (1988).
G. E. Collins, Quantifier elimination for real closed fields by cylindrical algebraic decomposition. Lect. Notes in Comp. Sci., 33, 515–532. Springer-Verlag (1975).
M. Coste,Effective semi-algebraic geometry. Lect. Notes in Comp. Sci., 391,1–27. Springer-Verlag (1989).
[12] L. Gonzalez-Vega, F. Rouillier, M.-F. Roy, Symbolic recipes for polynomial system solving. In: Some tapas for computer algebra, A. Cohen ed.Springer, 34–65 (1999).
[13] L. Gonzalez-Vega, F. Rouillier, M.-F. Roy, G. Trujillo, Symbolic recipes for real solutions. In: Some tapas for computer algebra, A. Cohen ed. Springer, 121–167 (1999).
D. G rigor’ev, N. Vorobjov, Solving Systems of Polynomial Inequalities in Subexponential Time. J. Symbolic Comput., 5, 37–64, (1988).
J. Heintz, T. Recio, M.-F. Roy, Algorithms in real algebraic geometry and applications to computational geometry. Discrete and Computational Geometry:Papers from the DIMACS Special Year. DIMACS series in Discrete Mathematics and Theoretical Computer Science AMS-ACM, 6, 137–163 (1991).
J. Heintz, M.-F. Roy, P. Solerno, On the Complexity of Semi-Algebraic Sets. Proc. IFIP 89, San Francisco, 293–298. North-Holland (1989).
J. Milnor, Morse Theory, Annals of Mathematical Studies, Princeton University Press (1963).
J. Milnor, On the Betti numbers of real algebraic varieties. Proc. AMS, 15,275–280 (1964).
O. A. Oleinick, Estimates on the Betti numbers of real algebraic hypersurfaces.Mat. Sb., 28 (70), 635–640 (1951).
I. Petrowsky, On the topology of real plane algebraic curves. An. Math., 39(1), 187–209, (1938).
R. Pollack, M.-F. Roy, On the number of cells defined by a set of polynomials.C. R. Acad. Sci. Paris, 316, 573–577 (1993).
J. Renegar, On the computational complexity and geometry of the first order theory of the reals. J. of Symbolic Comput.,13 (3), 255–352 (1992).
J. Renegar, Recent progress on the complexity of the decision problem for the reals. Discrete and Computational Geometry: Papers from the DIMACS Special Year. DIMACS series in Discrete Mathematics and Theoretical Computer Science, AMS-ACM., 6, 287–308 (1991).
F. Rouillier, Algorithmes efficaces pour l’´etude des z´eros r´eels des syst`emes polynomiaux. Th`ese, Universit´e de Rennes I (1996).
F. Rouillier, Solving zero-dimensional systems through the rational univariate representation. Journal of Applicable Algebra in Engineering, Communication and Computing,9, 433–461 (1999).
F. Rouillier, Real Solving,http://www.loria.fr/rouillie/rouillie
M.-F. Roy Basic algorithms in real algebraic geometry: from Sturm theorem to the existential theory of reals,, Lectures on Real Geometry in memoriam of Mario Raimondo, de Gruyter Expositions in Mathematics, 1–67 (1996).
M.-F. Roy, N. Vorobjov, Computing the complexification of a semi-algebraic set, Math. Zeitschrift 239, 131–142 (2002).
M. Safey El din R´esolution r´eelle des syst`emes polynomiaux en dimension positive. Th`ese, Universit´e Paris VI (2001).
[30] R. Thom, Sur l’homologie des vari´et´es r´eelles. In: Differential and combinatorial topology, 255–265. Princeton University Press (1965).
[31] R. J. Walker, Algebraic Curves. Princeton University Press (1950).
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Roy, MF. (2003). Some Recent Quantitative and Algorithmic Results in Real Algebraic Geometry. In: Aronov, B., Basu, S., Pach, J., Sharir, M. (eds) Discrete and Computational Geometry. Algorithms and Combinatorics, vol 25. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-55566-4_34
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DOI: https://doi.org/10.1007/978-3-642-55566-4_34
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