AI summary · arXiv: 1209.5021v1 · math.CO
Canonical Forms of $2 \times 2 \times 2$ and $2 \times 2 \times 2 \times 2$ Symmetric Tensors over Prime Fields
The essentials
- The paper explores symmetric tensors of format $2 \times 2 \times 2$ and $2 \times 2 \times 2\times 2$ over prime fields.
- The main objective is to compute the canonical forms of these tensors using computer algebra techniques.
- For the $2\times2\times2$ case, prime fields $\mathbb{F}_p$, where $p = [1,3,5,7,11,13,17]$, are considered.
- For the $4\times4\times4\times4$ case, prime fields $\mathbb{F}_p$, where $p = [1,3,5,7]$, are taken into account.
- The author determines the canonical forms for each type of tensor using computer algebra techniques.
- Canonical forms provide a unique representation for each tensor in its equivalence class under certain transformations.
- This study investigates properties related to these canonical forms such as size of orbit and symmetric rank of tensors within its orbit.
- Size of an orbit refers to the number of distinct tensors that can be transformed into a given canonical form using symmetry-preserving operations.
- Symmetric rank measures how well a tensor can be approximated by lower-rank symmetric tensors.
- The findings contribute to a deeper understanding of symmetric tensors and their canonical forms with applications in areas such as algebraic geometry, representation theory and computer science.
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