Quasi-homogeneous polynomial

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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.

In other words, a polynomial is quasi-homogeneous 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 . 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 polytope of lies in the affine space with equation inside .

The above equation is equivalent to this new one: . Some authors[1] 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

whenever .


  1. ^ J. Steenbrink (1977). Compositio Mathematica, tome 34, n° 2. Noordhoff International Publishing. p. 211 (Available on-line at Numdam)