A characterization of semi-quasihomogeneous functions in terms of the Milnor number

We give an inequality on the Milnor number $\mu (f)$ of a hypersurface isolated singularity in terms of the weighted Taylor expansion $f = f_{\rho }+ f_{\rho +1} + \cdots$ for a given weight on the coordinates. Here the equality holds if and only if the leading term $f_{\rho }$ defines an isolated singularity. This gives a characterization of the semi-quasihomogeneous condition in terms of $\mu$. Our proof uses a result on multiplicity of filtered rings and is given by purely algebraic arguments.