Quasi-homogeneous polynomial

In algebra, a multivariate polynomial

[math]\displaystyle{ 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 quasi-homogeneous or weighted homogeneous, if there exist r integers [math]\displaystyle{ w_1, \ldots, w_r }[/math], called weights of the variables, such that the sum [math]\displaystyle{ w=w_1i_1+ \cdots + w_ri_r }[/math] 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 from the fact that a polynomial f is quasi-homogeneous if and only if

[math]\displaystyle{ f(\lambda^{w_1} x_1, \ldots, \lambda^{w_r} x_r)=\lambda^w f(x_1,\ldots, x_r) }[/math]

for every [math]\displaystyle{ \lambda }[/math] in any field containing the coefficients.

A polynomial [math]\displaystyle{ f(x_1, \ldots, x_n) }[/math] is quasi-homogeneous with weights [math]\displaystyle{ w_1, \ldots, w_r }[/math] if and only if

[math]\displaystyle{ f(y_1^{w_1}, \ldots, y_n^{w_n}) }[/math]

is a homogeneous polynomial in the [math]\displaystyle{ y_i }[/math]. 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 [math]\displaystyle{ \alpha }[/math] belong to the same affine hyperplane. As the Newton polytope of the polynomial is the convex hull of the set [math]\displaystyle{ \{\alpha \mid a_\alpha \neq0 \}, }[/math] 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 [math]\displaystyle{ f(x,y)=5x^3y^3+xy^9-2y^{12} }[/math], which is not homogeneous. However, if instead of considering [math]\displaystyle{ f(\lambda x, \lambda y) }[/math] we use the pair [math]\displaystyle{ (\lambda^3, \lambda) }[/math] to test homogeneity, then

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

The above equation is equivalent to this new one: [math]\displaystyle{ \tfrac{1}{4}x + \tfrac{1}{12}y = 1 }[/math]. Some authors prefer to use this last condition and prefer to say that our polynomial is quasi-homogeneous of type [math]\displaystyle{ (\tfrac{1}{4},\tfrac{1}{12}) }[/math].

As noted above, a homogeneous polynomial [math]\displaystyle{ 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]\displaystyle{ 1i_1+1i_2 = d }[/math].

Definition

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

[math]\displaystyle{ 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]\displaystyle{ \varphi=(\varphi_1,\ldots,\varphi_r) }[/math], [math]\displaystyle{ \varphi_i\in\mathbb{N} }[/math], if there exists some [math]\displaystyle{ a \in \mathbb{R} }[/math] such that

[math]\displaystyle{ \langle \alpha,\varphi \rangle = \sum_k^ri_k\varphi_k = a }[/math]

whenever [math]\displaystyle{ a_\alpha\neq 0 }[/math].