#jsDisabledContent { display:none; } My Account |  Register |  Help

# Quasi-homogeneous polynomial

Article Id: WHEBN0022236311
Reproduction Date:

 Title: Quasi-homogeneous polynomial Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

### Quasi-homogeneous polynomial

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},

is quasi-homogeneous or weighted homogeneous, if there exists r integers w_1, \ldots, w_r, called weights of the variables, such that the sum w=w_1i_1+ \cdots + w_ri_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(\lambda^{w_1} x_1, \ldots, \lambda^{w_r} x_r)=\lambda^w f(x_1,\ldots, x_r)

for every \lambda in any field containing the coefficients.

A polynomial f(x_1, \ldots, x_n) is quasi-homogeneous with weights w_1, \ldots, w_r if and only if

f(y_1^{w_1}, \ldots, 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 \alpha belong to the same affine hyperplane. As the Newton polygon of the polynomial is the convex hull of the set \{\alpha | a_\alpha\neq0\}, 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").

## Introduction

Consider the polynomial f(x,y)=5x^3y^3+xy^9-2y^{12}. This one has no chance of being a homogeneous polynomial; however if instead of considering f(\lambda x,\lambda y) we use the pair (\lambda^3,\lambda) to test homogeneity, then

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). \,

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 3i_1+1i_2=12. In particular, this says that the Newton polygon of f(x,y) lies in the affine space with equation 3x+y=12 inside \mathbb{R}^2.

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 d is just a quasi-homogeneous polynomial of type (1,1); in this case all its pairs of exponents will satisfy the equation 1i_1+1i_2=d.

## Definition

Let f(x) be a polynomial in r variables x=x_1\ldots x_r with coefficients in a commutative ring R. We express it as a finite sum

f(x)=\sum_{\alpha\in\mathbb{N}^r} a_\alpha x^\alpha, \alpha=(i_1,\ldots,i_r), a_\alpha\in \mathbb{R}.

We say that f is quasi-homogeneous of type \varphi=(\varphi_1,\ldots,\varphi_r), \varphi_i\in\mathbb{N} if there exists some a\in\mathbb{R} such that

\langle \alpha,\varphi \rangle = \sum_k^ri_k\varphi_k=a,

whenever a_\alpha\neq 0.

## References

1. ^ J. Steenbrink (1977). Compositio Mathematica, tome 34, n° 2. Noordhoff International Publishing. p. 211 (Available on-line at Numdam)
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.

Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.