On the Moduli Space of Quasi-Homogeneous Functions

Abstract

We relate the moduli space of analytic equivalent germs of reduced quasi-homogeneous functions at $$({\mathbb {C}}^{2},0)$$ with their bi-Lipschitz equivalence classes. We show that any non-degenerate continuous family of (reduced) quasi-homogeneous (but not homogeneous) functions with constant Henry–Parusiński invariant is analytically trivial. Further, we show that there are only a finite number of distinct bi-Lipschitz classes among quasi-homogeneous functions with the same Henry–Parusiński invariant providing a maximum quota for this number. Finally, we conclude that the moduli space of bi-Lipschitz equivalent quasi-homogeneous function-germs admits an analytic structure.