Quasi-homogeneous polynomial


In algebra, a multivariate polynomial
is quasi-homogeneous or weighted homogeneous, if there exist 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 from 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.
A polynomial is quasi-homogeneous if and only if all the 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.

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
, because its three pairs of exponents, and all satisfy the linear equation. In particular, this says that the Newton polytope of lies in the affine space with equation inside.
The above equation is equivalent to this new one:. Some authors 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 ; 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
whenever.