The symmetric rank and decomposition of m-order n-dimensional (n = 2,3,4) symmetric tensors over the binary field

Abstract

Firstly, we propose an algebraic method to compute the symmetric rank and symmetric rank decomposition for symmetric tensors over the binary field, when the order is at least the dimension minus one. Then we discuss Comon's conjecture and rank decomposition for low dimension symmetric tensors. For 2×2×⋯×2 symmetric tensors over the binary field, we show Comon's conjecture is true except for one special tensor (actually matrix), and the rank decomposition problem is solved when the order is greater than or equal to 4. For 3×3×⋯×3 symmetric tensors over the binary field, we prove Comon's conjecture is true when the order is greater than or equal to 4, and obtain the unique rank decomposition when the order is greater than or equal to 6. Finally, for 4×4×⋯×4 symmetric tensors over the binary field, we show Comon's conjecture is true and obtain the unique rank decomposition when the order is greater than or equal to 8.