Efficiently Computing Real Roots of Sparse Polynomials

Abstract

We propose an efficient algorithm to compute the real roots of a sparse polynomial f ∈ R[x] having k non-zero realvalued 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 ∆1, . . . , ∆s ⊂ C, with s < 2k, centered at the real axis and of radius less than 2 −L together with positive integers µ1, . . . , µs such that each disk ∆i contains exactly µi 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 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 log n, and near-linear in τ and log σ −1 , where σ denotes the separation of the real roots.

Condition Numbers for the Cube. I: Univariate Polynomials and Hypersurfaces

“…The latter would be ideal in view of the results of Khovanski ˘i [16] and Kushnirenko's hypothesis, which bound the size of the Betti numbers of zero sets of sparse polynomials independently of the degree. However, few progress have been made in this direction beyond the univariate case [15]. Moreover, many computational problems in real algebraic geometry lack algorithms that are polynomial in the degree, so such bounds contribute to the state-of-the-art.…”