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 \lambda in any field containing the coefficients. A polynomial is quasi-homogeneous with weights if and only if 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, which is not homogeneous. However, if instead of considering we use the pair to test homogeneity, then We say that 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. In particular, this says that the Newton polytope of f(x,y) lies in the affine space with equation 3x+y = 12 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 g(x,y) of degree d is just a quasi-homogeneous polynomial of type (1,1)

Definition

Let f(x) 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.