Homogeneity in generalized function algebras

We investigate homogeneity in the special Colombeau algebra on R^d as well as on the pierced space R^d {set minus} {0}. It is shown that strongly scaling invariant functions on R^d are simply the constants. On the pierced space, strongly homogeneous functions of degree α admit tempered representatives, whereas on the whole space, such functions are polynomials with generalized coefficients. We also introduce weak notions of homogeneity and show that these are consistent with the classical notion on the distributional level. Moreover, we investigate the relation between generalized solutions of the Euler differential equation and homogeneity.