Quasi-homogeneous polynomial


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.

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 (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. ^ Steenbrink, J. (1977). "Intersection form for quasi-homogeneous singularities" (PDF). Compositio Mathematica. 34 (2): 211–223 See p. 211. ISSN 0010-437X.