Orbits of monomials and factorization into products of linear forms

This paper is devoted to the factorization of multivariate polynomials into products of linear forms, a problem which has applications to differential algebra, to the resolution of systems of polynomial equations and to Waring decomposition (i.e., decomposition in sums of d-th powers of linear forms; this problem is also known as symmetric tensor decomposition). We provide three black box algorithms for this problem. Our main contribution is an algorithm motivated by the application to Waring decomposition. This algorithm reduces the corresponding factorization problem to simultaenous matrix diagonalization, a standard task in linear algebra. The algorithm relies on ideas from invariant theory, and more specifically on Lie algebras. Our second algorithm reconstructs a factorization from several bi-variate projections. Our third algorithm reconstructs it from the determination of the zero set of the input polynomial, which is a union of hyperplanes.