On computing the determinant in small parallel time using a small number of processors
Information Processing Letters
Parallel algorithms for algebraic problems
SIAM Journal on Computing
Efficient parallel algorithms
Generalised characteristic polynomials
Journal of Symbolic Computation - Special issue on computational algebraic complexity
On the complexity of diophantine geometry in low dimensions (extended abstract)
STOC '99 Proceedings of the thirty-first annual ACM symposium on Theory of computing
Counting points on curves and Abelian varieties over finite fields
Journal of Symbolic Computation
Complexity and expressive power of logic programming
ACM Computing Surveys (CSUR)
Complexity and Expressive Power of Logic Programming
CCC '97 Proceedings of the 12th Annual IEEE Conference on Computational Complexity
Electronic Notes in Theoretical Computer Science (ENTCS)
The multivariate resultant is NP-hard in any characteristic
MFCS'10 Proceedings of the 35th international conference on Mathematical foundations of computer science
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In this paper we develop a fast parallel procedure for deciding when a set of multivariate polynomials with coefficients in an arbitrary field K have a common algebraic solution. Moreover, since the proposed algorithm is algebraic, it easily yields a procedure for quantifier elimination in the theory of an arbitrary algebraically closed field.More precisely, we show how to decide whether m polynomials in n variables, each of degree at most d, with coefficients in an arbitrary field K have a common zero in the algebraic closure of K, using sequential time (mn)&Ogr;(n)d&Ogr;(n)2), or parallel time &Ogr;(n3 log3 d log m) with (mn)&Ogr;(n)d&Ogr;(n)2) processors, in the operations of the coefficient field K. Using randomization, this may be improved to (mn)&Ogr;(1)d&Ogr;(n) time.In addition, the construction is used give a direct EXPSPACE algorithm for quantifier elimination in the theory of an algebraically-closed field, which runs in PSPACE or parallel polynomial time when restricted to formulas with a fixed number of alternations of quantifiers.