Absolute reconstruction for sums of powers of linear forms: degree 3 and beyond

Abstract

We study the decomposition of multivariate polynomials as sums of powers of linear forms. We give a randomised algorithm for the following problem: If a homogeneous polynomial $f \in K[x_1,..., x_n]$ (where $K \subseteq \mathbb {C}$) 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:

• For $d = 3$, we improve by a factor of n on the running time from the algorithm in Koiran & Skomra (2021). The price to be paid for this improvement is that the algorithm now has two-sided error.

• For $d > 3$, we provide the first randomised blackbox algorithm for this problem that runs in time $\text {poly}(n,d)$ (in an algebraic model where only arithmetic operations and equality tests are allowed). Previous algorithms for this problem Kayal (2011) as well as most of the existing reconstruction algorithms for other classes appeal to a polynomial factorisation 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.

• For $d > 3$, when f has rational coefficients (i.e. $K = \mathbb {Q}$), 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 $\mathbb {C}$ with polynomial running time in the bit model of computation.

These results are true even when we replace $\mathbb {C}$ by $\mathbb {R}$. We view the problem as a tensor decomposition problem and use linear algebraic methods such as checking the simultaneous diagonalisability of the slices of a tensor. The number of such slices is exponential in d. But surprisingly, we show that after a random change of variables, computing just 3 special slices is enough. We also show that our approach can be extended to the computation of the actual decomposition. In forthcoming work we plan to extend these results to overcomplete decompositions, i.e. decompositions in more than n powers of linear forms.