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 exist r integers $w_{1},\ldots ,w_{r}$, 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

$f(\lambda ^{w_{1}}x_{1},\ldots ,\lambda ^{w_{r}}x_{r})=\lambda ^{w}f(x_{1},\ldots ,x_{r})$

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.

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 $\{\alpha \mid a_{\alpha }\neq 0\},$ 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 $f(x,y)=5x^{3}y^{3}+xy^{9}-2y^{12}$, 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: ${\tfrac {1}{4}}x+{\tfrac {1}{12}}y=1$. Some authors 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

$\langle \alpha ,\varphi \rangle =\sum _{k}^{r}i_{k}\varphi _{k}=a$

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