Quasi-homogeneous polynomial explained

In algebra, a multivariate polynomial

f(x)=\sum\alphaa\alpha

\alpha,where\alpha=(i
x
1,...,i

r)\inNr,and

i1
x
1

ir
x
r

,

is quasi-homogeneous or weighted homogeneous, if there exist r integers

w1,\ldots,wr

, called weights of the variables, such that the sum

w=w1i1++wrir

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

w1
f(λ

x1,\ldots,

wr
λ
w
x
r)

f(x1,\ldots,xr)

for every

λ

in any field containing the coefficients.

A polynomial

f(x1,\ldots,xn)

is quasi-homogeneous with weights

w1,\ldots,wr

if and only if
w1
f(y
1

,\ldots,

wn
y
n

)

is a homogeneous polynomial in the

yi

. 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

\alpha

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

f(x,y)=5x3y3+xy9-2y12

, which is not homogeneous. However, if instead of considering

f(λx,λy)

we use the pair

(λ3,λ)

to test homogeneity, then

f(λ3x,λy)=5(λ3x)3(λy)3+(λ3x)(λy)9-2(λy)12=λ12f(x,y).

We say that

f(x,y)

is a quasi-homogeneous polynomial of type, because its three pairs of exponents, and all satisfy the linear equation

3i1+1i2=12

. In particular, this says that the Newton polytope of

f(x,y)

lies in the affine space with equation

3x+y=12

inside

R2

.

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

g(x,y)

of degree is just a quasi-homogeneous polynomial of type ; in this case all its pairs of exponents will satisfy the equation

1i1+1i2=d

.

Definition

Let

f(x)

be a polynomial in variables

x=x1\ldotsxr

with coefficients in a commutative ring . We express it as a finite sum
f(x)=\sum
\alpha\inNr

a\alphax\alpha,\alpha=(i1,\ldots,ir),a\alpha\inR.

We say that is quasi-homogeneous of type

\varphi=(\varphi1,\ldots,\varphir)

,

\varphii\inN

, if there exists some

a\inR

such that

\langle\alpha,\varphi\rangle=

ri
\sum
k\varphi

k=a

whenever

a\alpha0

.

Notes and References

  1. J. . Steenbrink . Intersection form for quasi-homogeneous singularities . Compositio Mathematica . 34 . 2 . 211–223 See p. 211 . 1977 . 0010-437X .