2011/12/19 by Samir Datta, Datta, Samir, Rameshwar Pratap +1
Computer Science · Mathematics · #Benford’s Law and Fraud Detection #Computability, Logic, AI Algorithms #Computational Complexity (cs.CC) #FOS: Computer and information sciences #Numerical Methods and Algorithms #cs.CC
paper · pdf · doi:10.48550/arxiv.1112.4295
arxiv created 2011/12/19 · openalex publication_date 2011/12/19 · arxiv updated 2011/12/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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⊆ Logspace 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.