Quasi-homogeneous polynomial

In algebra, a multivariate polynomial
is quasi-homogeneous or weighted homogeneous, if there exist "r" integers formula_2, called weights of the variables, such that the sum formula_3 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 formula_5 in any field containing the coefficients. 
A polynomial formula_6 is quasi-homogeneous with weights formula_2 if and only if 
is a homogeneous polynomial in the formula_9. 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 formula_10 belong to the same affine hyperplane. As the Newton polytope of the polynomial is the convex hull of the set formula_11 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 formula_12, which is not homogeneous. However if instead of considering formula_13 we use the pair formula_14 to test homogeneity, then
We say that formula_16 is a quasi-homogeneous polynomial of type
, because its three pairs of exponents , and all satisfy the linear equation formula_17. In particular, this says that the Newton polytope of formula_16 lies in the affine space with equation formula_19 inside formula_20.
The above equation is equivalent to this new one: formula_21. Some authors prefer to use this last condition and prefer to say that our polynomial is quasi-homogeneous of type formula_22.
As noted above, a homogeneous polynomial formula_23 of degree is just a quasi-homogeneous polynomial of type ; in this case all its pairs of exponents will satisfy the equation formula_24.
Definition.
Let formula_25 be a polynomial in variables formula_26 with coefficients in a commutative ring . We express it as a finite sum
We say that is quasi-homogeneous of type formula_28, formula_29, if there exists some formula_30 such that 
whenever formula_32.