Quasi-homogeneous polynomial

In algebra, a multivariate polynomial
: $f(x)=\sum_\alpha a_\alpha x^\alpha\text\alpha=(i_1,\dots,i_r)\in \mathbb^r \text x^\alpha=x_1^ \cdots x_r^,$
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 . This sum is the ''weight'' or the ''degree'' of the polynomial.

The term ''quasi-homogeneous'' comes from the fact that a polynomial is quasi-homogeneous if and only if
:$f(\lambda^ x_1, \ldots, \lambda^ x_r)=\lambda^w f(x_1,\ldots, x_r)$
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
:$f(y_1^, \ldots, y_n^)$
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 $\,$ 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 $f(x,y)=5x^3y^3+xy^9-2y^$, 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

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

We say that $f(x,y)$ is a quasi-homogeneous polynomial of type
, because its three pairs of exponents , and 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 $\mathbb^2$.

The above equation is equivalent to this new one: $\tfracx + \tfracy = 1$. Some authors prefer to use this last condition and prefer to say that our polynomial is quasi-homogeneous of type $(\tfrac,\tfrac)$.

As noted above, a homogeneous polynomial $g(x,y)$ of degree is just a quasi-homogeneous polynomial of type ; 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 variables $x=x_1\ldots x_r$ with coefficients in a commutative ring . We express it as a finite sum

: $f(x)=\sum_ a_\alpha x^\alpha, \alpha=(i_1,\ldots,i_r), a_\alpha\in \mathbb.$

We say that is quasi-homogeneous of type $\varphi=(\varphi_1,\ldots,\varphi_r)$, $\varphi_i\in\mathbb$, if there exists some $a \in \mathbb$ such that

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

whenever $a_\alpha\neq 0$.

References

{{Polynomials

Commutative algebra
Algebraic geometry