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 $$w=w_1i_1+ \cdots + w_ri_r$$ 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 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 $$\{\alpha \mid a_\alpha \neq0 \},$$ 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 $$f(x,y)=5x^3y^3+xy^9-2y^{12}$$, 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


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

We say that $$f(x,y)$$ is a quasi-homogeneous polynomial of type $(3,1)$, because its three pairs $(i_{1}, i_{2})$ 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


 * $$\langle \alpha,\varphi \rangle = \sum_k^ri_k\varphi_k = a$$

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