For a homogeneous polynomial 
of polynomial
degree 
on a finite-dimensional vector space 
, the 
th catalecticant is the tensor
contraction map
 |
where 
.
Here 
denotes the vector space of symmetric
tensors of rank 
built from 
,
and 
is the corresponding space built from the dual space 
. The map contracts 
vector factors against 
. In monomial bases,
this map is represented by a catalecticant matrix.
The ranks and minors of
catalecticant matrices give conditions on
representations of forms as sums of powers.
