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Polynomial factorization and nonrandomness of bits of algebraic and some transcendental numbers. (English) Zbl 0654.12001
The authors apply the lattice basis reduction method of A. K. Lenstra, H. W. Lenstra jun. and L. Lovász [Math. Ann. 261, 515–534 (1982; Zbl 0488.12001)] to determine the minimal polynomial \(m_{\alpha}(t)\) of an algebraic number \(\alpha\), provided that they know bounds for the degree and the coefficients of \(m_{\alpha}\) and, depending on those bounds, sufficiently many digits of the real and imaginary part of \(\alpha\).
The running time of their algorithm is polynomial in the input data. The algorithm can of course be applied to the factorization of polynomials over the rationals. The authors also discuss the randomness of the bit sequence of the binary expansion of an algebraic number \(\alpha\) (and of some transcendental numbers like \(\log(\alpha)\), \(\cos^{-1}(\alpha))\) on the basis of that algorithm.
Reviewer: M.Pohst

11R09 Polynomials (irreducibility, etc.)
11Y16 Number-theoretic algorithms; complexity
11Y40 Algebraic number theory computations
68W30 Symbolic computation and algebraic computation
Full Text: DOI
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