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
Factoring rational polynomials over the complex numbers
SIAM Journal on Computing
Complexity of the Wu-Ritt decomposition
PASCO '97 Proceedings of the second international symposium on Parallel symbolic computation
The space complexity of elimination theory: upper bounds
FoCM '97 Selected papers of a conference on Foundations of computational mathematics
Computing an equidimensional decomposition of an algebraic variety by means of geometric resolutions
ISSAC '00 Proceedings of the 2000 international symposium on Symbolic and algebraic computation
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
Parallel algorithms for algebraic problems
STOC '83 Proceedings of the fifteenth annual ACM symposium on Theory of computing
The Computational Complexity of the Chow Form
Foundations of Computational Mathematics
Lifting and recombination techniques for absolute factorization
Journal of Complexity
Differential forms in computational algebraic geometry
Proceedings of the 2007 international symposium on Symbolic and algebraic computation
On the complexity of counting components of algebraic varieties
Journal of Symbolic Computation
Counting Irreducible Components of Complex Algebraic Varieties
Computational Complexity
Effective de Rham cohomology: the hypersurface case
Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation
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We prove two versions of Stickelberger's Theorem for positive dimensions and use them to compute the connected and irreducible components of a complex algebraic variety. If the variety is given by polynomials of degree @?d in n variables, then our algorithms run in parallel (sequential) time (nlogd)^O^(^1^) (d^O^(^n^^^4^)). In the case of a hypersurface, the complexity drops to O(n^2log^2d) (d^O^(^n^)). In the proof of the last result we use the effective Nullstellensatz for two polynomials, which we also prove by very elementary methods.