Power Sum Decompositions of Elementary Symmetric Polynomials

TL;DR

This paper investigates the tensor ranks of elementary symmetric polynomials, providing explicit power sum decompositions and establishing bounds that are tight for odd degrees.

Contribution

It offers new bounds on tensor ranks and explicit decompositions for elementary symmetric polynomials, advancing understanding in algebraic complexity.

Findings

Tensor rank bounds for elementary symmetric polynomials

Explicit power sum decompositions provided

Bounds are tight for odd degrees

Abstract

We bound the tensor ranks of elementary symmetric polynomials, and we give explicit decompositions into powers of linear forms. The bound is attained when the degree is odd.