Black Box Absolute Reconstruction for Sums of Powers of Linear Forms

TL;DR

This paper introduces a randomized blackbox algorithm for decomposing multivariate polynomials into sums of powers of linear forms, improving efficiency for degree 3 and providing the first polynomial-time algorithm for higher degrees over complex numbers.

Contribution

The paper presents novel randomized algorithms for polynomial decomposition, notably improving degree 3 case and establishing the first polynomial-time method for higher degrees over complex numbers.

Findings

Improved degree 3 decomposition algorithm with faster runtime but two-sided error.

First polynomial-time randomized algorithm for degrees > 3 over complex numbers.

Algorithm's runtime is polynomial in input size for rational coefficients.

Abstract

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 $f \in K[x_1 , . . . , x_n]$ (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: (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 $\mathbb{C}$ with polynomial running time in the bit model of computation.