Arthur-Merlin games: a randomized proof system, and a hierarchy of complexity class
Journal of Computer and System Sciences - 17th Annual ACM Symposium in the Theory of Computing, May 6-8, 1985
Solving systems of polynomial inequalities in subexponential time
Journal of Symbolic Computation
Complexity of deciding Tarski algebra
Journal of Symbolic Computation
Some algebraic and geometric computations in PSPACE
STOC '88 Proceedings of the twentieth annual ACM symposium on Theory of computing
PP is as hard as the polynomial-time hierarchy
SIAM Journal on Computing
Journal of Symbolic Computation
On the intrinsic complexity of elimination theory
Journal of Complexity - Special issue: invited articles dedicated to J. F. Traub on the occasion of his 60th birthday
Counting connected components of a semialgebraic set in subexponential time
Computational Complexity
Computing over the reals with addition and order
Selected papers of the workshop on Continuous algorithms and complexity
Algebraic decision trees and Euler characteristics
Theoretical Computer Science
Computing over the reals with addition and order: higher complexity classes
Journal of Complexity
On real Turing machines that toss coins
STOC '95 Proceedings of the twenty-seventh annual ACM symposium on Theory of computing
Hilbert's Nullstellensatz is in the polynomial hierarchy
Journal of Complexity - Special issue for the Foundations of Computational Mathematics conference, Rio de Janeiro, Brazil, Jan. 1997
On the Power of Real Turing Machines over Binary Inputs
SIAM Journal on Computing
A weak version of the Blum, Shub, and Smale model
Journal of Computer and System Sciences - Special issue: dedicated to the memory of Paris Kanellakis
Complexity and real computation
Complexity and real computation
Some complexity results for polynomial ideals
Journal of Complexity
Elimination of constants from machines over algebraically closed fields
Journal of Complexity
Elimination of parameters in the polynomial hierarchy
Theoretical Computer Science
Journal of Symbolic Computation
The real dimension problem is NPR -complete
Journal of Complexity
Counting problems over the reals
Theoretical Computer Science
Cook's versus Valiant's hypothesis
Theoretical Computer Science - Selected papers in honor of Manuel Blum
Constraint Databases
Randomized and deterministic algorithms for the dimension of algebraic varieties
FOCS '97 Proceedings of the 38th Annual Symposium on Foundations of Computer Science
Completeness classes in algebra
STOC '79 Proceedings of the eleventh annual ACM symposium on Theory of computing
Counting Complexity Classes for Numeric Computations I: Semilinear Sets
SIAM Journal on Computing
Counting Complexity Classes for Numeric Computations. III: Complex Projective Sets
Foundations of Computational Mathematics
Algorithms in Real Algebraic Geometry (Algorithms and Computation in Mathematics)
Algorithms in Real Algebraic Geometry (Algorithms and Computation in Mathematics)
Complexity of the mover's problem and generalizations
SFCS '79 Proceedings of the 20th Annual Symposium on Foundations of Computer Science
On a generalization of Stickelberger's Theorem
Journal of Symbolic Computation
Optimization and approximation problems related to polynomial system solving
CiE'06 Proceedings of the Second conference on Computability in Europe: logical Approaches to Computational Barriers
Exotic quantifiers, complexity classes, and complete problems
ICALP'07 Proceedings of the 34th international conference on Automata, Languages and Programming
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We define counting classes #P"R and #P"C in the Blum-Shub-Smale setting of computations over the real or complex numbers, respectively. The problems of counting the number of solutions of systems of polynomial inequalities over R, or of systems of polynomial equalities overC, respectively, turn out to be natural complete problems in these classes. We investigate to what extent the new counting classes capture the complexity of computing basic topological invariants of semialgebraic sets (over R) and algebraic sets (over C). We prove that the problem of computing the Euler-Yao characteristic of semialgebraic sets is FP"R^#^P^"^R-complete, and that the problem of computing the geometric degree of complex algebraic sets is FP"C^#^P^"^C-complete. We also define new counting complexity classes in the classical Turing model via taking Boolean parts of the classes above, and show that the problems to compute the Euler characteristic and the geometric degree of (semi)algebraic sets given by integer polynomials are complete in these classes. We complement the results in the Turing model by proving, for all k@?N, the FPSPACE-hardness of the problem of computing the kth Betti number of the set of real zeros of a given integer polynomial. This holds with respect to the singular homology as well as for the Borel-Moore homology.