Methods and Applications of Analysis

Volume 24 (2017)

Number 2

Special issue dedicated to Henry B. Laufer on the occasion of his 70th birthday: Part 2

Guest Editors: Stephen S.-T. Yau (Tsinghua University, China); Gert-Martin Greuel (University of Kaiserslautern, Germany); Jonathan Wahl (University of North Carolina, USA); Rong Du (East China Normal University, China); Yun Gao (Shanghai Jiao Tong University, China); and Huaiqing Zuo (Tsinghua University, China)

Zero-dimensional gradient singularities

Pages: 169 – 184



A. G. Aleksandrov (Institute for Control Sciences RAS, Moscow, Russia)

H.-Q. Zuo (Yau Mathematical Sciences Center, Tsinghua University, Beijing, China)


We discuss an approach to the problem of classifying zero-dimensional gradient quasihomogeneous singularities using simple properties of deformation theory. As an example, we enumerate all such singularities with modularity $\mathscr{P} = 0$ and with Milnor number not greater than $12$. We also compute normal forms and monomial vector-bases of the first cotangent homology and cohomology modules, the corresponding Poincaré polynomials, inner modality, inner modularity, primitive ideals, etc.


gradient singularities, multiple points, deformations, cotangent homology and cohomology, primitive ideals, complete intersections, inner modality and modularity

2010 Mathematics Subject Classification

14F10, 14F40, 32S25, 58K45, 58K70

Received 11 November 2016

Accepted 11 September 2017

Published 3 January 2018