Symmetric Tensor Decompositions over Finite Fields

TL;DR

This paper investigates symmetric tensor ranks in finite fields using linearized polynomials, providing explicit criteria, recovering known complexities, and introducing new invariants related to rank-metric codes.

Contribution

It introduces a novel approach to symmetric tensor decompositions over finite fields via linearized polynomials and applies it to compute complexities and define new invariants.

Findings

Recovered known symmetric bilinear complexities for small degrees

Derived explicit symmetric decompositions for specific parameters

Established the symmetric tensor rank of a Gabidulin code as an invariant

Abstract

We study the symmetric tensor rank of multiplication over finite field extensions using linearized polynomials. Via field trace, symmetric linearized polynomials are identified with symmetric bilinear forms and symmetric matrices, allowing symmetric tensor decompositions to be reformulated as spanning problems by rank-one symmetric linearized polynomials. We translate these spanning conditions into explicit linear systems over finite fields and use the Frobenius automorphism to obtain computationally effective criteria. As applications, we recover known values of the symmetric bilinear complexity for small extension degrees and obtain explicit symmetric decompositions for several parameters. We also introduce the symmetric tensor-rank of a symmetric rank-metric code and show that, for the natural one-dimensional Gabidulin code associated with finite field multiplication, this invariant coincides with the symmetric tensor rank of the multiplication map.