Eigenvectors of tensors and algorithms for Waring decomposition

TL;DR

This paper explores algorithms for computing Waring decompositions of polynomials, linking tensor eigenvectors and secant varieties, and explicitly decomposes a general cubic polynomial in three variables.

Contribution

It introduces new algorithms for Waring decomposition, connecting tensor eigenvectors with secant varieties, and provides an explicit decomposition of a cubic polynomial in three variables.

Findings

Algorithms for Waring decomposition are developed and analyzed.

Explicit decomposition of a general cubic polynomial in three variables as five cubes.

Connections established between tensor eigenvectors, secant varieties, and polynomial decompositions.

Abstract

A Waring decomposition of a (homogeneous) polynomial f is a minimal sum of powers of linear forms expressing f. Under certain conditions, such a decomposition is unique. We discuss some algorithms to compute the Waring decomposition, which are linked to the equation of certain secant varieties and to eigenvectors of tensors. In particular we explicitly decompose a general cubic polynomial in three variables as the sum of five cubes (Sylvester Pentahedral Theorem).