is a polynomial, then there exists r integers , called weights of the variables such that the sum is the same for all terms of f. This sum is called the weight or the degree of the polynomial. In other words, the convex hull of the set lies entirely on an affine hyperplane.
The term quasi-homogeneous comes form the fact that a polynomial f is quasi-homogeneous if and only if
for every in the field of the coefficients. A homogeneous polynomial is quasi-homogeneous for all weights equal to 1.
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 (i1,i2) 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 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 .
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
- J. Steenbrink (1977). Compositio Mathematica, tome 34, n° 2. Noordhoff International Publishing. p. 211 (Available on-line at Numdam)