On computing the determinant in small parallel time using a small number of processors
Information Processing Letters
Fast parallel absolute irreducibility testing
Journal of Symbolic Computation
Parallel arithmetic computations: a survey
Proceedings of the 12th symposium on Mathematical foundations of computer science 1986
Basic principles of mechanical theorem proving in elementary geometrics
Journal of Automated Reasoning
Some algebraic and geometric computations in PSPACE
STOC '88 Proceedings of the twentieth annual ACM symposium on Theory of computing
Gröbner bases and primary decomposition of polynomial ideals
Journal of Symbolic Computation
Factoring rational polynomials over the complexes
ISSAC '89 Proceedings of the ACM-SIGSAM 1989 international symposium on Symbolic and algebraic computation
A new method for solving algebraic systems of positive dimension
Discrete Applied Mathematics - Special volume on applied algebra, algebraic algorithms, and error-correcting codes
A generalized Euclidean algorithm for computing triangular representations of algebraic varieties
Journal of Symbolic Computation
Irreducible decomposition of algebraic varieties via characteristic sets and Gro¨bner bases
Computer Aided Geometric Design
Prime decompositions of radicals in polynomial rings
Journal of Symbolic Computation
Complexity of the Wu-Ritt decomposition
PASCO '97 Proceedings of the second international symposium on Parallel symbolic computation
Some complexity results for polynomial ideals
Journal of Complexity
The space complexity of elimination theory: upper bounds
FoCM '97 Selected papers of a conference on Foundations of computational mathematics
Algorithmic properties of polynomial rings
Journal of Symbolic Computation
On the theories of triangular sets
Journal of Symbolic Computation - Special issue on polynomial elimination—algorithms and applications
Absolute Primality of Polynomials is Decidable in Random Polynomial Time in the Number of Variables
Proceedings of the 8th Colloquium on Automata, Languages and Programming
Factoring multivariate polynomials via partial differential equations
Mathematics of Computation
Counting Complexity Classes for Numeric Computations. III: Complex Projective Sets
Foundations of Computational Mathematics
The complexity of semilinear problems in succinct representation
Computational Complexity
Algorithms in Real Algebraic Geometry (Algorithms and Computation in Mathematics)
Algorithms in Real Algebraic Geometry (Algorithms and Computation in Mathematics)
On the complexity of counting components of algebraic varieties
Journal of Symbolic Computation
On a generalization of Stickelberger's Theorem
Journal of Symbolic Computation
A note on Gao's algorithm for polynomial factorization
Theoretical Computer Science
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We give a uniform method for the two problems #CCC and #ICC of counting connected and irreducible components of complex algebraic varieties, respectively. Our algorithms are purely algebraic, i.e., they use only the field structure of C. They work efficiently in parallel and can be implemented by algebraic circuits of polynomial depth, i.e., in parallel polynomial time. The design of our algorithms relies on the concept of algebraic differential forms. A further important building block is an algorithm of Sántó [40] computing a variant of characteristic sets. The crucial complexity parameter for #ICC turns out to be the number of equations. We describe a randomised algorithm solving #ICC for a fixed number of rational equations given by straight-line programs (slps), which runs in parallel polylogarithmic time in the length and the degree of the slps.