Symmetric Tensor Decompositions over Finite Fields

2026-05-12

AI Summary

This study investigates the symmetric tensor rank of multiplication maps over finite field extensions. By leveraging linearized polynomials, the field trace, and the Frobenius automorphism, the problem of symmetric tensor decomposition is reformulated as one of generating such tensors via rank-one symmetric linearized polynomials. A computable criterion based on symmetric bilinear forms is established through a linear system framework. The work establishes, for the first time, a connection between symmetric tensor rank and symmetric rank-metric codes, introduces new invariants for symmetric tensor rank, and provides explicit symmetric decompositions for finite field multiplication. The proposed method successfully reproduces known symmetric bilinear complexities for small extension degrees and yields novel explicit decompositions for several parameter settings.

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.

Problem

• symmetric tensor rank
• finite fields
• multiplication map
• symmetric bilinear complexity
• rank-metric code

Innovation

• symmetric tensor decomposition
• linearized polynomials
• finite fields
• symmetric bilinear complexity
• rank-metric codes