Efficiently Computing Real Roots of Sparse Polynomials
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We propose an efficient algorithm to compute the real roots of a sparse polynomial f∈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 Δ_1,...,Δ_s⊂C, with s<2k, centered at the real axis and of radius less than 2^-L together with positive integers μ_1,...,μ_s such that each disk Δ_i contains exactly μ_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 n, and near-linear in L and τ, where 2^-τ and 2^τ 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 n, and near-linear in τ and σ^-1, where σ denotes the separation of the real roots.
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