The symmetric tensor rank of a symmetric tensor 
of order 
over 
or 
is the least 
for which
 |
where 
and the vectors 
belong to its underlying vector
space. The notation 
means the 
-fold tensor direct product
of 
with itself. Here 
and 
are the real and complex
numbers, respectively (Comon et al. 2008).
Symmetric tensor rank is at least tensor decomposition rank, since requiring the factors in each summand to coincide restricts the allowed tensor decompositions. The Comon conjecture asserted equality, but counterexamples are known. For a symmetric tensor of order 2, both ranks equal its matrix rank.
