Quasi-homogeneous polynomial

In algebra, a multivariate polynomial

${\displaystyle f(x)=\sum _{\alpha }a_{\alpha }x^{\alpha }{\text{, where }}\alpha =(i_{1},\dots ,i_{r})\in \mathbb {N} ^{r}{\text{, and }}x^{\alpha }=x_{1}^{i_{1}}\cdots x_{r}^{i_{r}},}$

is quasi-homogeneous or weighted homogeneous, if there exist r integers ${\displaystyle w_{1},\ldots ,w_{r}}$, called weights of the variables, such that the sum ${\displaystyle w=w_{1}i_{1}+\cdots +w_{r}i_{r}}$ 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

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

for every ${\displaystyle \lambda }$ in any field containing the coefficients.

A polynomial ${\displaystyle f(x_{1},\ldots ,x_{n})}$ is quasi-homogeneous with weights ${\displaystyle w_{1},\ldots ,w_{r}}$ if and only if

${\displaystyle f(y_{1}^{w_{1}},\ldots ,y_{n}^{w_{n}})}$

is a homogeneous polynomial in the ${\displaystyle y_{i}}$. 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 ${\displaystyle \alpha }$ belong to the same affine hyperplane. As the Newton polytope of the polynomial is the convex hull of the set ${\displaystyle \{\alpha \mid a_{\alpha }\neq 0\},}$ 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 ${\displaystyle f(x,y)=5x^{3}y^{3}+xy^{9}-2y^{12}}$, which is not homogeneous. However, if instead of considering ${\displaystyle f(\lambda x,\lambda y)}$ we use the pair ${\displaystyle (\lambda ^{3},\lambda )}$ to test homogeneity, then

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

We say that ${\displaystyle f(x,y)}$ 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 ${\displaystyle 3i_{1}+1i_{2}=12}$. In particular, this says that the Newton polytope of ${\displaystyle f(x,y)}$ lies in the affine space with equation ${\displaystyle 3x+y=12}$ inside ${\displaystyle \mathbb {R} ^{2}}$.

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

As noted above, a homogeneous polynomial ${\displaystyle g(x,y)}$ 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 ${\displaystyle 1i_{1}+1i_{2}=d}$.

Definition

Let ${\displaystyle f(x)}$ be a polynomial in r variables ${\displaystyle x=x_{1}\ldots x_{r}}$ with coefficients in a commutative ring R. We express it as a finite sum

${\displaystyle f(x)=\sum _{\alpha \in \mathbb {N} ^{r}}a_{\alpha }x^{\alpha },\alpha =(i_{1},\ldots ,i_{r}),a_{\alpha }\in \mathbb {R} .}$

We say that f is quasi-homogeneous of type ${\displaystyle \varphi =(\varphi _{1},\ldots ,\varphi _{r})}$, ${\displaystyle \varphi _{i}\in \mathbb {N} }$, if there exists some ${\displaystyle a\in \mathbb {R} }$ such that

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

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

References