Maximal multihomogeneity of algebraic hypersurface singularities

From the degree zero part of the logarithmic vector fields along analgebraic hypersurface singularity we identify the maximal multihomogeneity of a defining equation in form of a maximal algebraic torus in the embedded automorphism group. We show that all such maximal tori are conjugate and in one–to–one correspondence to maximal tori in the linear jet of the embedded automorphism group. These results are motivated by Kyoji Saito’s characterization of quasihomogeneity for isolated hypersurface singularities [Saito in Invent. Math. 14, 123–142 (1971)] and extend previous work with Granger and Schulze [Compos. Math. 142(3), 765–778 (2006), Theorem 5.4] and of Hauser and Müller [Nagoya Math. J. 113, 181–186 (1989), Theorem 4].