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Efficient Algorithms for Sparse Cyclotomic Integer Zero Testing

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Abstract

We present two deterministic polynomial time algorithms for the following problem: check whether a sparse polynomial f(x) vanishes at a given primitive nth root of unity ζ n . A priori f(ζ n ) may be nonzero and doubly exponentially small in the input size. The existence of a polynomial time procedure in the case of factored n was conjectured by D. Plaisted in 1984, but all previously known algorithms are either randomized, or do not run in polynomial time.

We apply polynomial zero testing algorithms to construct a nondeterministic polynomial time algorithm for the torsion point problem (TP). The problem TP is a particular case of the feasibility problem for a system of polynomial equations in complex numbers (coefficients of polynomials are integers). In the problem TP all coordinates of a solution must be roots of unity.

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Authors and Affiliations

  1. School of Computer Science, The University of Oklahoma, Norman, OK, 73019, USA

    Qi Cheng

  2. Dorodnitsyn Computing Center of RAS, Vavilova, 40, Moscow, 119991, Russia

    Sergey P. Tarasov & Mikhail N. Vyalyi

Authors
  1. Qi Cheng
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  2. Sergey P. Tarasov
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  3. Mikhail N. Vyalyi
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Corresponding author

Correspondence to Mikhail N. Vyalyi.

Additional information

The research of Q. Cheng was partially supported by NSF Career Award CCR-0237845 of USA and by Project 973 (no: 2007CB807903) of China.

The research of S.P. Tarasov was supported by the RFBR grant 08–01–00414.

The research of M.N. Vyalyi was supported by the RFBR grants 08–01–00414, 05–01–02803–NTsNIL_a and the grant NS 5294.2008.1.

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Cheng, Q., Tarasov, S.P. & Vyalyi, M.N. Efficient Algorithms for Sparse Cyclotomic Integer Zero Testing. Theory Comput Syst 46, 120–142 (2010). https://doi.org/10.1007/s00224-008-9158-2

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