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48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07) (2007)
Providence, Rhode Island
Oct. 21, 2007 to Oct. 23, 2007
ISSN: 0272-5428
ISBN: 0-7695-3010-9
pp: 74-80
ABSTRACT
The zero testing and sign determination problems of real algebraic numbers of high extension degree are important in computational complexity and numerical analysis. In this paper we concentrate on sparse cyclotomic integers. Given an integer n and a sparse polynomial f(x) = ckx^{ek} + ck - 1x^{ek - 1} + \cdot \cdot \cdot + c1x^{e1} over {\rm Z}, we present a deterministic polynomial time algorithm to decide whether f(\omega n) is zero or not, where \omega n denotes the n-th primitive root of unity e^{2\pi \sqrt { - 1/n} }. All previously known algorithms are either randomized, or do not run in polynomial time. As a side result, we prove that if n is free of prime factors less than k + 1, there exist k field automorphisms \sigma _{1,} \sigma _{2,} \cdot \cdot \cdot _, \sigma _k in the Galois group Gal {\text{(Q(}}\omega {\text{n)/Q)}} such that for any nonzero integers c_{1,} c_{2,} \cdot \cdot \cdot _, c_k and for any integers 0 \leqslant e1 \le e2 \le \cdots \le ek \le n, there exists i so that \left| {\sigma i} \right.(ck\omega _n^{ek} + ck - 1\omega _n^{ek - 1} + \cdots + c_1^{} \omega _n^{e1} \left. ) \right| \geqslant 1/2^{(k^2 \log n + k\log k)} .
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CITATION

Q. Cheng, "Derandomization of Sparse Cyclotomic Integer Zero Testing," 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)(FOCS), Providence, Rhode Island, 2007, pp. 74-80.
doi:10.1109/FOCS.2007.15
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