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
.
