Efficiently Computing Real Roots of Sparse Polynomials

We propose an efficient algorithm to compute the real roots of a sparse polynomial f is an element of R[x] having k non-zero real-valued coefficients. It is assumed that arbitrarily good approximations of the non-zero coefficients are given by means of a coefficient oracle. For a given positive integer L, our algorithm returns disjoint disks Delta(1),..., Delta(s) subset of C, with s < 2k, centered at the real axis and of radius less than 2(-L) together with positive integers mu(1),..., mu(s) such that each disk Delta(i) contains exactly mu(i) roots of f counted with multiplicity. In addition, it is ensured that each real root of f is contained in one of the disks. If f has only simple real roots, our algorithm can also be used to isolate all real roots. The bit complexity of our algorithm is polynomial in k and log n, and near-linear in L and tau, where 2(-tau) and 2(tau) constitute lower and upper bounds on the absolute values of the non-zero coefficients of f, and n is the degree of f. For root isolation, the bit complexity is polynomial in k and log n, and near-linear in tau and log sigma(-1), where sigma denotes the separation of the real roots.