# Black Box Absolute Reconstruction for Sums of Powers of Linear Forms

We study the decomposition of multivariate polynomials as sums of powers of linear forms. We give a randomized algorithm for the following problem: If a homogeneous polynomial f ∈ K[x_1 , . . . , x_n] (where K ⊆ℂ) of degree d is given as a blackbox, decide whether it can be written as a linear combination of d-th powers of linearly independent complex linear forms. The main novel features of the algorithm are: (1) For d = 3, we improve by a factor of n on the running time from an algorithm by Koiran and Skomra. The price to be paid for this improvement though is that the algorithm now has two-sided error. (2) For d > 3, we provide the first randomized blackbox algorithm for this problem that runs in time polynomial in n and d (in an algebraic model where only arithmetic operations and equality tests are allowed). Previous algorithms for this problem as well as most of the existing reconstruction algorithms for other classes appeal to a polynomial factorization subroutine. This requires extraction of complex polynomial roots at unit cost and in standard models such as the unit-cost RAM or the Turing machine this approach does not yield polynomial time algorithms. (3) For d > 3, when f has rational coefficients, the running time of the blackbox algorithm is polynomial in n,d and the maximal bit size of any coefficient of f. This yields the first algorithm for this problem over ℂ with polynomial running time in the bit model of computation.

READ FULL TEXT