In algebra, a multivariate polynomial is quasi-homogeneous or weighted homogeneous, if there exist r integers
, called weights of the variables, such that the sum
is the same for all nonzero terms of f. This sum w is the weight or the degree of the polynomial.
The term quasi-homogeneous comes from the fact that a polynomial f is quasi-homogeneous if and only if for every
λ
in any field containing the coefficients.
is quasi-homogeneous with weights
if and only if is a homogeneous polynomial in the
In particular, a homogeneous polynomial is always quasi-homogeneous, with all weights equal to 1.
A polynomial is quasi-homogeneous if and only if all the
α
belong to the same affine hyperplane.
As the Newton polytope of the polynomial is the convex hull of the set
{ α ∣
the quasi-homogeneous polynomials may also be defined as the polynomials that have a degenerate Newton polytope (here "degenerate" means "contained in some affine hyperplane").
to test homogeneity, then We say that
is a quasi-homogeneous polynomial of type (3,1), because its three pairs (i1, i2) of exponents (3,3), (1,9) and (0,12) all satisfy the linear equation
In particular, this says that the Newton polytope of
lies in the affine space with equation
inside
The above equation is equivalent to this new one:
Some authors[1] prefer to use this last condition and prefer to say that our polynomial is quasi-homogeneous of type
As noted above, a homogeneous polynomial
of degree d is just a quasi-homogeneous polynomial of type (1,1); in this case all its pairs of exponents will satisfy the equation
be a polynomial in r variables
with coefficients in a commutative ring R. We express it as a finite sum We say that f is quasi-homogeneous of type
φ = (
φ
φ
φ