Efficient Algorithms for Sparse Cyclotomic Integer Zero Testing

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.