Dagstuhl - Seminar ( 9545 ) on Real Computation and Complexity

On the Complexity of Real Equation Solving by Bernd Bank, Humboldt–Universität zu Berlin The first part of the speech deals with the particular case of integer points in a semi-algebraic set described by quasi-convex polynomial inequalities in the n–dimensional real space. The simply exponential bounds are “optimal” with respect to the dense codification of the polynomials. Without changing the data structure there is no hope to improve the complexity. In a second part it is discussed, what can be expected if the method recently developed by GIUSTI/ HEINTZ/ MORAIS/ MORGENSTERN/ PARDO is transferred to the real case. This method finds the isolated points in a zero–dimensional affine variety. Its main features are the use of straight line programs as data structure and the polynomial sequential time measured in both, the length of the input description and an appropriate affine “geometric” degree of the equation system. The main result with respect to the real case is the following: the transferred method finds a representative point with algebraic coordinates of each connected component of a given smooth and compact real hypersurface. A suitably defined “real degree” of some polar variety corresponding to the input equation describing the hypersurface in question replaces the affine geometric degree of the equation system. On Bounding the Betti Numbers of Semi-Algebraic Sets by Saugata Basu, Courant Institute, New York University In this talk we give a new bound on the sum of the Betti numbers of semialgebraic sets. This extends a well-known bound due to Oleinik and Petrovsky, Thom and Milnor. In separate papers they proved that the sum of the Betti numbers of a semi-algebraic set S ⊂ R, defined by P1 ≥ 0, . . . , Ps ≥ 0, deg(Pi) ≤ d, 1 ≤ i ≤ s, is bounded by (O(sd)). Given a semi-algebraic set S ⊂ R defined