A fast parallel algorithm for determining all roots of a polynomial with real roots
SIAM Journal on Computing
Threshold circuits of small majority-depth
Information and Computation
Fundamental problems of algorithmic algebra
Fundamental problems of algorithmic algebra
Introduction to Circuit Complexity: A Uniform Approach
Introduction to Circuit Complexity: A Uniform Approach
Uniform constant-depth threshold circuits for division and iterated multiplication
Journal of Computer and System Sciences - Complexity 2001
Computational Complexity: A Modern Approach
Computational Complexity: A Modern Approach
On the Complexity of Numerical Analysis
SIAM Journal on Computing
| Hi-index | 0.00 |
We initiate the complexity theoretic study of the problem of computing the bits of (real) algebraic numbers. This extends the work of Yap on computing the bits of transcendental numbers like π , in Logspace. Our main result is that computing a bit of a fixed real algebraic number is in C= NC1 $\subseteq \mbox{{\sf L}}$ when the bit position has a verbose (unary) representation and in the counting hierarchy when it has a succinct (binary) representation. Our tools are drawn from elementary analysis and numerical analysis, and include the Newton-Raphson method. The proof of our main result is entirely elementary, preferring to use the elementary Liouville's theorem over the much deeper Roth's theorem for algebraic numbers. We leave the possibility of proving non-trivial lower bounds for the problem of computing the bits of an algebraic number given the bit position in binary, as our main open question. In this direction we show very limited progress by proving a lower bound for rationals .