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A '''quasi-homogeneous polynomial''' is a [[polynomial]] which has a degenerate [[Newton polygon]]. This means that if
: <math>f(x)=\sum_\alpha a_\alpha x^\alpha\text{, where }\alpha=(i_1,\dots,i_r)\in \mathbb{N}^r \text{, and } x^\alpha=x_1^{i_1} \cdots x_r^{i_r}</math>
is a polynomial, then there ''r'' integers <math>w_1, \ldots, w_r</math>, called '''weights''' of the variables such that the sum <math>w=w_1i_1+ \cdots + w_ri_r</math> is the same for all terms of ''f''. This sum is called the ''weight'' or the ''degree'' of the polynomial. In other words, the [[convex hull]] of the set <math>\{\alpha | a_\alpha\neq0\}</math> lies entirely on an affine hyperplane.


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The term ''quasi-homogeneous'' comes form the fact that a polynomial ''f'' is quasi-homogeneous if and only if
:<math> f(\lambda^{w_1} x_1, \ldots, \lambda^{w_r} x_r)=\lambda^w f(x_1,\ldots, x_r)</math>
for every <math>\lambda</math>  in the field of the coefficients. A [[homogeneous polynomial]] is quasi-homogeneous for all weights equal to 1.
 
==Introduction==
Consider the polynomial <math>f(x,y)=5x^3y^3+xy^9-2y^{12}</math>. This one has no chance of being a [[homogeneous polynomial]]; however if instead of considering <math>f(\lambda x,\lambda y)</math> we use the pair <math>(\lambda^3,\lambda)</math> to test ''homogeneity'', then
 
: <math>f(\lambda^3 x,\lambda y)=5(\lambda^3x)^3(\lambda y)^3+(\lambda^3x)(\lambda y)^9-2(\lambda y)^{12}=\lambda^{12}f(x,y). \, </math>  
 
We say that <math>f(x,y)</math> is a quasi-homogeneous polynomial of '''type'''
(3,1), because its three pairs (''i''<sub>1</sub>,''i''<sub>2</sub>) of exponents (3,3), (1,9) and (0,12) all satisfy the linear equation <math>3i_1+1i_2=12</math>. In particular, this says that the Newton polygon of <math>f(x,y)</math> lies in the affine space with equation <math>3x+y=12</math> inside <math>\mathbb{R}^2</math>.
 
The above equation is equivalent to this new one: <math>\tfrac{1}{4}x+\tfrac{1}{12}y=1</math>. Some authors<ref>J. Steenbrink (1977). ''Compositio Mathematica'', tome 34, n° 2. Noordhoff International Publishing. p. 211 (Available on-line at [http://www.numdam.org/ Numdam])</ref> prefer to use this last condition and prefer to say that our polynomial is quasi-homogeneous of type (<math>\tfrac{1}{4},\tfrac{1}{12}</math>).
 
As noted above, a homogeneous polynomial <math>g(x,y)</math> 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 <math>1i_1+1i_2=d</math>.
 
==Definition==
Let <math>f(x)</math> be a polynomial in ''r'' variables <math>x=x_1\ldots x_r</math> with coefficients in a commutative ring ''R''. We express it as a finite sum
 
: <math>f(x)=\sum_{\alpha\in\mathbb{N}^r} a_\alpha x^\alpha, \alpha=(i_1,\ldots,i_r), a_\alpha\in \mathbb{R}.</math>
 
We say that ''f'' is '''quasi-homogeneous of type''' <math>\varphi=(\varphi_1,\ldots,\varphi_r)</math>, <math>\varphi_i\in\mathbb{N}</math> if there exists some <math>a\in\mathbb{R}</math> such that
 
: <math>\langle \alpha,\varphi \rangle = \sum_k^ri_k\varphi_k=a,</math>
 
whenever <math>a_\alpha\neq 0</math>.
 
==References==
<references/>
 
[[Category:Commutative algebra]]
[[Category:Algebraic geometry]]

Revision as of 02:00, 30 December 2013

A quasi-homogeneous polynomial is a polynomial which has a degenerate Newton polygon. This means that if

f(x)=αaαxα, where α=(i1,,ir)r, and xα=x1i1xrir

is a polynomial, then there r integers w1,,wr, called weights of the variables such that the sum w=w1i1++wrir is the same for all terms of f. This sum is called the weight or the degree of the polynomial. In other words, the convex hull of the set {α|aα0} lies entirely on an affine hyperplane.

The term quasi-homogeneous comes form the fact that a polynomial f is quasi-homogeneous if and only if

f(λw1x1,,λwrxr)=λwf(x1,,xr)

for every λ in the field of the coefficients. A homogeneous polynomial is quasi-homogeneous for all weights equal to 1.

Introduction

Consider the polynomial f(x,y)=5x3y3+xy92y12. This one has no chance of being a homogeneous polynomial; 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)92(λy)12=λ12f(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 3i1+1i2=12. In particular, this says that the Newton polygon of f(x,y) lies in the affine space with equation 3x+y=12 inside 2.

The above equation is equivalent to this new one: 14x+112y=1. Some authors[1] prefer to use this last condition and prefer to say that our polynomial is quasi-homogeneous of type (14,112).

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 1i1+1i2=d.

Definition

Let f(x) be a polynomial in r variables x=x1xr with coefficients in a commutative ring R. We express it as a finite sum

f(x)=αraαxα,α=(i1,,ir),aα.

We say that f is quasi-homogeneous of type φ=(φ1,,φr), φi if there exists some a such that

α,φ=krikφk=a,

whenever aα0.

References

  1. J. Steenbrink (1977). Compositio Mathematica, tome 34, n° 2. Noordhoff International Publishing. p. 211 (Available on-line at Numdam)