The weighted degree of a monomial with respect to assigned positive integer weights 
is the sum of its exponents
multiplied by the corresponding weights. For nonnegative
integer exponents 
,
 |
The weighted degree of a nonzero polynomial is the largest weighted degree of its nonzero monomials (Faugère et al. 2016). Here the weighted degree of the zero polynomial is left undefined.
When all weights are 1, weighted degree agrees with total degree. A weighted homogeneous
polynomial has the same weighted degree in every nonzero monomial. For example, with weights 2 for 
and 3 for 
, the polynomial 
has weighted degree 6, whereas its total
degree is 3.
