Derandomization of Sparse Cyclotomic Integer Zero Testing

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 an sparse cyclotomic integers. Given an integer n and a sparse polynomial f(x) = C_kx^e(k) + c_k-1x^e(k-1) + ... + c₁x^e(1)over Z, we present a deterministic polynomial time algorithm to decide whether f(w_n) is zero or not, where f(w_n) denotes the n-th primitive root of unity e^{2piradic(-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 sigma₁, sigma₂, ... , sigma_k in the Galois group Gal (Q(w_n)/Q) such that for any nonzero integers c₁, c₂ ... , c_k and for any integers 0 les e₁ < e₂ < ... < e_k < n, there exists i so that |sigma_i(c_kw_n ^ek + c_k-1w_n ^e(k-1) + ... + c₁w_n ^e(1)) | ges 1/2^{(k(2)log n+klogk)}.