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
Fast parallel absolute irreducibility testing
Journal of Symbolic Computation
Representations and parallel computations for rational functions
SIAM Journal on Computing
Complexity of parallel matrix computations
Theoretical Computer Science
A fast parallel algorithm for determining all roots of a polynomial with real roots
SIAM Journal on Computing
Some algebraic and geometric computations in PSPACE
STOC '88 Proceedings of the twentieth annual ACM symposium on Theory of computing
On the worst-case arithmetic complexity of approximating zeros of systems of polynomials
SIAM Journal on Computing
Parallel evaluation of the determinant and of the inverse of a matrix
Information Processing Letters
Simple algorithms for approximating all roots of a polynomial with real roots
Journal of Complexity
Subresultants and Reduced Polynomial Remainder Sequences
Journal of the ACM (JACM)
The complexity of elementary algebra and geometry
STOC '84 Proceedings of the sixteenth annual ACM symposium on Theory of computing
SFCS '85 Proceedings of the 26th Annual Symposium on Foundations of Computer Science
A faster PSPACE algorithm for deciding the existential theory of the reals
SFCS '88 Proceedings of the 29th Annual Symposium on Foundations of Computer Science
STOC '95 Proceedings of the twenty-seventh annual ACM symposium on Theory of computing
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
Univariate polynomials: nearly optimal algorithms for factorization and rootfinding
Proceedings of the 2001 international symposium on Symbolic and algebraic computation
Univariate polynomials: nearly optimal algorithms for numerical factorization and root-finding
Journal of Symbolic Computation - Computer algebra: Selected papers from ISSAC 2001
Analysis of Asynchronous Polynomial Root Finding Methods on a Distributed Memory Multicomputer
IEEE Transactions on Parallel and Distributed Systems
FST TCS '02 Proceedings of the 22nd Conference Kanpur on Foundations of Software Technology and Theoretical Computer Science
Proceedings of the forty-second ACM symposium on Theory of computing
Advice coins for classical and quantum computation
ICALP'11 Proceedings of the 38th international colloquim conference on Automata, languages and programming - Volume Part I
Journal of the ACM (JACM)
On some complexity issues of NC analytic functions
TAMC'06 Proceedings of the Third international conference on Theory and Applications of Models of Computation
Root finding with threshold circuits
Theoretical Computer Science
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Given a polynomial p(z) of degree n with integer coefficients, whose absolute values are bounded above by 2^m, and a specified integer @m, we show that the problem of determining all roots of p with error less than 2^-^@m is in the parallel complexity class NC. To do this, we construct an algorithm which runs on at most D(n + m + @m)^f processors in at most C log^e(n + m - @m) parallel steps, where the constants C, D, e, f are given in terms of the corresponding processor and time bounds for the computation of certain elementary polynomial and matrix operations. In fact, one can easily see that the time complexity is O(log^3(n + m + @m)). Thus, the algorithm presented here extends the algorithm of Ben-Or, Feig, Kozen, and Tiwari by removing the severe restriction that all the roots of p(z) be real.