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)