The initial form of a polynomial with respect to a real weight vector is the sum
of its polynomial terms of greatest weight. For
a nonzero polynomial 
over a field 
, let 
and put 
, where 
, 
is an exponent vector,
and the dot denotes the dot product. Then
 |
The initial form of the zero polynomial is defined to be zero. Several polynomial terms can have the same greatest weight, so an initial form need not be a single polynomial term (Fukuda et al. 2007).
For example, for 
over 
, the weights 
, 
, and 
give initial forms 
, 
,
and 
, respectively. Weight zero gives
. With respect to a monomial
order, the initial form instead means the leading
term. The least-weight convention is obtained by replacing 
by 
.
