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Univariate real root isolation over a single logarithmic extension of real algebraic numbers. (English) Zbl 1393.11086
Kotsireas, Ilias S. (ed.) et al., Applications of computer algebra, Kalamata, Greece, July 20–23, 2015. Cham: Springer (ISBN 978-3-319-56930-7/hbk; 978-3-319-56932-1/ebook). Springer Proceedings in Mathematics & Statistics 198, 425-445 (2017).
Summary: We present algorithmic, complexity, and implementation results for the problem of isolating the real roots of a univariate polynomial $$B \in L[x]$$, where $$L=\mathbb {Q} [ \lg (\alpha)]$$ and $$\alpha$$ is a positive real algebraic number. The algorithm approximates the coefficients of $$B$$ up to a sufficient accuracy and then solves the approximate polynomial. For this we derive worst-case (aggregate) separation bounds. We also estimate the expected number of real roots when we draw the coefficients from a specific distribution and illustrate our results experimentally. A generalization to bivariate polynomial systems is also presented. We implemented the algorithm in C as part of the core library of mathematica for the case $$B \in \mathbb {Z} [ \lg (q)][x]$$ where $$q$$ is positive rational number and we demonstrate its efficiency over various data sets.
For the entire collection see [Zbl 1379.13001].
##### MSC:
 11Y16 Number-theoretic algorithms; complexity 11R04 Algebraic numbers; rings of algebraic integers 12D10 Polynomials in real and complex fields: location of zeros (algebraic theorems) 68W30 Symbolic computation and algebraic computation
##### Software:
ISOLATE; Mathematica
Full Text:
##### References:
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