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Quasi-homogeneous polynomial

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Quasi-homogeneous polynomial

In algebra, a multivariate 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)
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