A weighted homogeneous polynomial is a polynomial for which every nonzero monomial
has the same weighted degree. For positive
integer weights 
and a nonnegative
integer 
, weighted homogeneity of degree 
is equivalent to the formal polynomial
identity
 |
where 
are independent variables. The zero
polynomial satisfies this identity for every 
.
When 
, the normalized weights are the positive rational numbers 
. For example, 
has weights 2 for 
and 3 for 
,
with weighted degree 6. Its monomials
have different ordinary total degrees. Equal weights
recover ordinary homogeneity, so the polynomial is
a homogeneous polynomial.
