Faster Algorithms via Waring Decompositions

07/17/2018

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by   Kevin Pratt, et al.

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We show that decompositions of certain polynomials as sums of powers of linear forms yield faster algorithms for some algebraic problems. Such a decomposition is known as a Waring decomposition. Our results are: 1. We give a black-box algorithm for computing the sum of the coefficients of the degree-k multilinear monomials in a polynomial over a field of characteristic zero. Our algorithm runs in time O^*(n^k/2) and space poly(n). This solves an open problem of Koutis and Williams [1]. 2. We give a randomized O^*((3e/2)^k) ≈ O^*(4.08^k) time, polynomial space, black-box algorithm for detecting multilinear monomials of degree k over a field of characteristic zero. This improves on the O^*(4.32^k) time, exponential space algorithm given in [2]. We note that there is a O^*(2^k) lower bound on our approach.