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 . 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
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 (here "degenerate" means "contained in some affine hyperplane").
Consider the polynomial , which is not homogeneous. 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 is just a quasi-homogeneous polynomial of type ; in this case all its pairs of exponents will satisfy the equation .
Let be a polynomial in variables with coefficients in a commutative ring . We express it as a finite sum
We say that is quasi-homogeneous of type , , if there exists some such that
whenever .