Quasi-homogeneous polynomial

In algebra, a multivariate polynomial

$f(x)=\sum _{\alpha }{a}_{\alpha }{x}^{\alpha }\text{, where }\alpha =(i_1,\dots ,i_r)\in {\mathbb{\mathbb{N}}}^r\text{, and }{x}^{\alpha }=x_1^{i_1}\cdots x_r^{i_r},$

is quasi-homogeneous or weighted homogeneous, if there exist r integers $w_1,\dots ,w_r$, called weights of the variables, such that the sum $w=w_1{i}_1+\cdots +w_r{i}_r$ 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 from the fact that a polynomial f is quasi-homogeneous if and only if

$f({\lambda }^{w_1}{x}_1,\dots ,{\lambda }^{w_r}{x}_r)={\lambda }^{w}f(x_1,\dots ,x_r)$

for every $\lambda$ in any field containing the coefficients. A polynomial $f(x_1,\dots ,x_n)$ is quasi-homogeneous with weights $w_1,\dots ,w_r$ if and only if

$f(y_1^{w_1},\dots ,y_n^{w_n})$

is a homogeneous polynomial in the $y_i$. In particular, a homogeneous polynomial is always quasi-homogeneous, with all weights equal to 1. A polynomial is quasi-homogeneous if and only if all the $\alpha$ belong to the same affine hyperplane. As the Newton polytope of the polynomial is the convex hull of the set $\{\alpha \mid a_{\alpha }\ne 0\},$ 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 $f(x,y)=5x^3{y}^3+x{y}^9-2y^{12}$, which is not homogeneous. However, if instead of considering $f(\lambda x,\lambda y)$ we use the pair $({\lambda }^3,\lambda )$ to test homogeneity, then

$f({\lambda }^{3}x,\lambda y)=5({\lambda }^{3}x{)}^3(\lambda y{)}^3+({\lambda }^{3}x)(\lambda y{)}^9-2(\lambda y{)}^{12}={\lambda }^{12}f(x,y).$

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 polytope 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: $\frac{1}{4}x+\frac{1}{12}y=1$. Some authors prefer to use this last condition and prefer to say that our polynomial is quasi-homogeneous of type $(\frac{1}{4},\frac{1}{12})$. 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 $x=x_1\dots x_r$ with coefficients in a commutative ring R. We express it as a finite sum

$f(x)=\sum _{\alpha \in {\mathbb{\mathbb{N}}}^r}{a}_{\alpha }{x}^{\alpha },\alpha =(i_1,\dots ,i_r),a_{\alpha }\in \mathbb{ℝ}.$

We say that f is quasi-homogeneous of type $\phi =({\phi }_1,\dots ,{\phi }_r)$, ${\phi }_i\in \mathbb{\mathbb{N}}$, if there exists some $a\in \mathbb{ℝ}$ such that

$⟨\alpha ,\phi ⟩=\sum _k^r{i}_k{\phi }_k=a$

whenever $a_{\alpha }\ne 0$.