In algebra, a multivariate polynomial
f ( x ) = ∑ α a α x α , where α = ( i 1 , … , i r ) ∈ N r , and x α = x 1 i 1 ⋯ x r i r , is quasi-homogeneous or weighted homogeneous, if there exists r integers w 1 , … , w r , called weights of the variables, such that the sum w = w 1 i 1 + ⋯ + w r i r 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 form the fact that a polynomial f is quasi-homogeneous if and only if
f ( λ w 1 x 1 , … , λ w r x r ) = λ w f ( x 1 , … , x r ) for every λ in any field containing the coefficients.
A polynomial f ( x 1 , … , x n ) is quasi-homogeneous with weights w 1 , … , w r if and only if
f ( y 1 w 1 , … , y n w n ) is a homogeneous polynomial in the y i . In particular, a homogeneous polynomial is always quasi-homogeneous, with all weights equal to 1.
In other words, a polynomial is quasi-homogeneous if all the α belong to the same affine hyperplane. As the Newton polygon of the polynomial is the convex hull of the set { α | a α ≠ 0 } , the quasi-homogeneous polynomials may also be defined as the polynomials that have a degenerate Newton polynomial (here "degenerate" means "contained in some affine hyperplane").
Consider the polynomial f ( x , y ) = 5 x 3 y 3 + x y 9 − 2 y 12 . This one has no chance of being a homogeneous polynomial; however if instead of considering f ( λ x , λ y ) we use the pair ( λ 3 , λ ) to test homogeneity, then
f ( λ 3 x , λ y ) = 5 ( λ 3 x ) 3 ( λ y ) 3 + ( λ 3 x ) ( λ y ) 9 − 2 ( λ y ) 12 = λ 12 f ( x , y ) . We say that f ( x , y ) 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 3 i 1 + 1 i 2 = 12 . In particular, this says that the Newton polygon of f ( x , y ) lies in the affine space with equation 3 x + y = 12 inside R 2 .
The above equation is equivalent to this new one: 1 4 x + 1 12 y = 1 . Some authors prefer to use this last condition and prefer to say that our polynomial is quasi-homogeneous of type ( 1 4 , 1 12 ).
As noted above, a homogeneous polynomial g ( x , y ) 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 1 i 1 + 1 i 2 = d .
Let f ( x ) be a polynomial in r variables x = x 1 … x r with coefficients in a commutative ring R. We express it as a finite sum
f ( x ) = ∑ α ∈ N r a α x α , α = ( i 1 , … , i r ) , a α ∈ R . We say that f is quasi-homogeneous of type φ = ( φ 1 , … , φ r ) , φ i ∈ N if there exists some a ∈ R such that
⟨ α , φ ⟩ = ∑ k r i k φ k = a , whenever a α ≠ 0 .
