# Quasi-homogeneous polynomial

In algebra, a multivariate polynomial

is **quasi-homogeneous** or **weighted homogeneous**, if there exists *r* integers , called **weights** of the variables, such that the sum 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 form the fact that a polynomial *f* is quasi-homogeneous if and only if

for every in any field containing the coefficients.

A polynomial is quasi-homogeneous with weights if and only if

is a homogeneous polynomial in the . In particular, a homogeneous polynomial is always quasi-homogeneous, with all weights equal to 1.

In other words, a polynomial is quasi-homogeneous if all the belong to the same affine hyperplane. As the Newton polygon 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 polynomial (here "degenerate" means "contained in some affine hyperplane").

## Introduction

Consider the polynomial . This one has no chance of being a homogeneous polynomial; however if instead of considering we use the pair to test *homogeneity*, then

We say that 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 . In particular, this says that the Newton polygon of lies in the affine space with equation inside .

The above equation is equivalent to this new one: . Some authors^{[1]} prefer to use this last condition and prefer to say that our polynomial is quasi-homogeneous of type ().

As noted above, a homogeneous polynomial 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 .

## Definition

Let be a polynomial in *r* variables with coefficients in a commutative ring *R*. We express it as a finite sum

We say that *f* is **quasi-homogeneous of type** , if there exists some such that