Abstract
We present a polynomial time algorithm for the following problem: to check whether a lacunary 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. Only exponential algorithms were known for this problem. The existence of an efficient procedure in the case of factored n was conjectured by D. Plaisted in 1984. As a consequence we show that the problem of the divisibility testing of a lacunary polynomial by some cyclotomic polynomial belongs to the complexity class \(\mathcal{NP}\).
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Tarasov, S.P., Vyalyi, M.N. (2007). An Efficient Algorithm for Zero-Testing of a Lacunary Polynomial at the Roots of Unity. In: Diekert, V., Volkov, M.V., Voronkov, A. (eds) Computer Science – Theory and Applications. CSR 2007. Lecture Notes in Computer Science, vol 4649. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74510-5_40
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DOI: https://doi.org/10.1007/978-3-540-74510-5_40
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