On Comon's conjecture over arbitrary fields

Abstract

Comon's Conjecture asserts that for a symmetric tensor, the rank is equal to the symmetric rank. For the field of real numbers R or the field of complex numbers C, Zhang et al. proved the conjecture for the case that the rank of a symmetric tensor does not exceed the order. In this paper, we show that over arbitrary fields this conjecture is true when the rank of a tensor is less than its order. Moreover, we construct a counterexample in which the rank of a symmetric tensor whose rank equals its order.