An efficient algorithm for zero-testing of a lacunary polynomial at the roots of unity

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 ζn. A priori f(ζn) 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 NP. © Springer-Verlag Berlin Heidelberg 2007.