Structural complexity 1
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
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
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
Hilbert''s Nullstellensatz is in the Polynomial Hierarchy
Hilbert''s Nullstellensatz is in the Polynomial Hierarchy
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)
Algebraic Complexity Theory
Differential forms in computational algebraic geometry
Proceedings of the 2007 international symposium on Symbolic and algebraic computation
A numerical algorithm for zero counting, I: Complexity and accuracy
Journal of Complexity
Kolmogorov Complexity Theory over the Reals
Electronic Notes in Theoretical Computer Science (ENTCS)
On the complexity of counting components of algebraic varieties
Journal of Symbolic Computation
Complexity of some geometric and topological problems
GD'09 Proceedings of the 17th international conference on Graph Drawing
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We define counting classes #PR and #PC 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 over C, 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 FPR#PR-complete, and that the problem of computing the geometric degree of complex algebraic sets is FPC#PC-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.