Quasi-Homogeneous Polynomial

(Algebraic Geometry)


Algebraic Geometry
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What is 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").

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algebraic geometrycommutative algebra


Weighted homogeneous polynomial

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