Quasihomogeneous

Quasihomogeneous (also spelled quasi-homogeneous) describes functions or polynomials that are homogeneous with respect to a weighted scaling of the variables. A function f: R^n → R (or C) is quasihomogeneous of degree d with weights w1, ..., wn > 0 if for every t > 0, f(t^{w1} x1, ..., t^{ wn} xn) = t^d f(x1, ..., xn). In the polynomial case this means every monomial x1^{α1} ... xn^{αn} in f has weighted degree α1 w1 + ... + αn wn = d. If all weights are equal, one recovers ordinary homogeneity.

Equivalently, for differentiable f one has sum_i w_i x_i ∂f/∂x_i = d f. This is the weighted version of Euler’s theorem on homogeneous functions.

A simple example is f(x, y) = x^2 + y^3, which is quasihomogeneous of degree 6 with weights (3, 2), since 2·3 = 6 and 3·2 = 6. Another example is a monomial x1^{α1} ... xn^{αn} which has quasihomogeneous degree Σ α_i w_i under the given weights.

In algebraic geometry and singularity theory, quasihomogeneous polynomials define cones and are used to study weighted projective spaces and the behavior of singularities under scaling. They also appear in the analysis of scaling symmetries in differential equations and dynamical systems. The term conveys a generalization of homogeneity by allowing different coordinates to scale with different weights.