Canonical Forms of $2 \times 2 \times 2$ and $2 \times 2 \times 2 \times 2$ Symmetric Tensors over Prime Fields

TL;DR

This paper computes canonical forms of symmetric tensors of sizes 2x2x2 and 2x2x2x2 over various prime fields using computer algebra, analyzing their orbits and symmetric ranks.

Contribution

It provides the first systematic computation of canonical forms and orbit sizes of these symmetric tensors over multiple prime fields.

Findings

Canonical forms for tensors over specified prime fields

Orbit sizes associated with each canonical form

Symmetric ranks of tensors in each orbit

Abstract

We consider symmetric tensors of format $2 \times 2 \times 2$ and $2 \times 2 \times 2 \times 2$ over prime fields. Using computer algebra we compute the canonical forms of these tensors. For $2 \times 2 \times 2$ symmetric tensors, we consider the prime fields $\mathbb{F}_p$ for $p = 2, 3, 5, 7, 11, 13, 17$. For $2 \times 2 \times 2 \times 2$ symmetric tensors, we consider the prime fields $\mathbb{F}_p$ for $p = 2, 3, 5, 7$. For each canonical form, we determine the size of its orbit and the symmetric rank of the symmetric tensors in its orbit.