Nine characterizations of weighted homogeneous isolated hypersurface singularities (Q1692610)

\(f\in \mathbb{C}[z_1, \ldots, z_n]\) is called weighted homogeneous if there exist positive rational numbers \(w_1, \ldots, w_n\) and \(d\) such that for each monomial \(\prod z_i^{a_i}\) appearing in \(f\) with nonzero coefficient one has \(\sum a_iw_i=d\). The polynomial \(f\) is called quasi-homogeneous if \(f\in J(f):=\langle\frac{\partial f}{\partial z_1}, \ldots, \frac{\partial f}{\partial z_n}\rangle\). Let \(V:=V(f)\) be the zero--set of \(f\), \(A(V):=\mathbb{C}[[z_1, \ldots, z_n]]/\langle f, J(f)\rangle\) the moduli algebra, \(M(V):=\mathbb{C}[[z_1, \ldots, z_n]]/J(f)\) the Milnor algebra, \(\mu=\dim_{\mathbb{C}} M(V)\) the Milnor number and \(\tau=\dim A(V)\) the Tjurina number. The paper gives nine characterizations of weighted homogeneous hypersurface singularities. The first characterization is the theorem of \textit{K. Saito} [Invent. Math. 23, 289--325 (1974; Zbl 0296.14019)] by \(\mu=\tau\). The second characterization by \textit{Y.-J. Xu} and \textit{S. S. T. Yau} [Am. J. Math. 118, No. 2, 389--399 (1996; Zbl 0927.32022)] is given by the property of the moduli algebra \(A(V)=\bigoplus_{i\geq 0}A_i\;,\;A_0=\mathbb{C}\), being a nonnegatively graded algebra. The third characterization by Xu and Yau is given in terms of the Lie-algebra of \(A(V)\). The following characterizations are in terms of geometric genus and irregularity [the authors, Commun. Anal. Geom. 21, No. 3, 509--526 (2013; Zbl 1302.32027)], in terms of \(D\)-module theory [\textit{S. Tajima} and \textit{Y. Nakamura}, Publ. Res. Inst. Math. Sci. 41, No. 1, 1--10 (2005; Zbl 1105.32020)], in terms of \(b\)-function theory [\textit{T. Yano}, Publ. Res. Inst. Math. Sci. 14, 111--202 (1978; Zbl 0389.32005)], in terms of module theory [\textit{A. Martsinkovsky}, Proc. Am. Math. Soc. 112, No. 1, 9--18 (1991; Zbl 0724.13022); \textit{J. Herzog}, Arch. Math. 59, No. 6, 556--561 (1992; Zbl 0811.14002)] and in terms of Jacobian syzygies [\textit{A. Dimca} and \textit{G. Sticlaru}, J. Symb. Comput. 74, 627--634 (2016; Zbl 1333.14037)].