Quasi-homogeneous polynomial explained
In algebra, a multivariate polynomial
f(x)=\sum\alphaa\alpha
| \alpha,where\alpha=(i |
x | |
| 1,...,i |
r)\inNr,and
…
,
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 . This sum is the
weight or the
degree of the polynomial.
The term quasi-homogeneous comes from the fact that a polynomial is quasi-homogeneous if and only if
x1,\ldots,
f(x1,\ldots,xr)
for every
in any
field containing the coefficients.
A polynomial
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
belong to the same affine hyperplane. As the
Newton polytope of the polynomial is the
convex hull of the set
\{\alpha\mida\alpha ≠ 0\},
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").
Introduction
Consider the polynomial
, which is not homogeneous. However, if instead of considering
we use the pair
to test homogeneity, then
f(λ3x,λy)=5(λ3x)3(λy)3+(λ3x)(λy)9-2(λy)12=λ12f(x,y).
We say that
is a quasi-homogeneous polynomial of
type, because its three pairs of exponents, and 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:
\tfrac{1}{4}x+\tfrac{1}{12}y=1
. Some authors
[1] prefer to use this last condition and prefer to say that our polynomial is quasi-homogeneous of type
(\tfrac{1}{4},\tfrac{1}{12})
.
As noted above, a homogeneous polynomial
of degree is just a quasi-homogeneous polynomial of type ; in this case all its pairs of exponents will satisfy the equation
.
Definition
Let
be a polynomial in variables
with coefficients in a
commutative ring . We express it as a finite sum
a\alphax\alpha,\alpha=(i1,\ldots,ir),a\alpha\inR.
We say that is quasi-homogeneous of type
\varphi=(\varphi1,\ldots,\varphir)
,
, if there exists some
such that
\langle\alpha,\varphi\rangle=
k=a
whenever
.
Notes and References
- J. . Steenbrink . Intersection form for quasi-homogeneous singularities . Compositio Mathematica . 34 . 2 . 211–223 See p. 211 . 1977 . 0010-437X .