By V.I. Arnol'd, J.S. Joel, V.V. Goryunov, O.V. Lyashko, V.A. Vasil'ev
In the 1st quantity of this survey (Arnol'd et al. (1988), hereafter stated as "EMS 6") we familiar the reader with the elemental thoughts and techniques of the speculation of singularities of delicate mappings and services. This idea has quite a few purposes in arithmetic and physics; the following we commence describing those applica tions. however the current quantity is largely self reliant of the 1st one: all the suggestions of singularity concept that we use are brought during the presentation, and references to EMS 6 are restricted to the quotation of technical effects. even supposing our major objective is the presentation of analready formulated thought, the readerwill additionally encounter a few relatively fresh effects, it appears unknown even to experts. We pointout a few of these effects. 2 three within the attention of mappings from C into C in§ three. 6 of bankruptcy 1, we outline the bifurcation diagram of one of these mapping, formulate a K(n, 1)-theorem for the enhances to the bifurcation diagrams of easy singularities, provide the definition of the Mond invariant N within the spirit of "hunting for invariants", and we draw the reader's cognizance to a mode of making a dead ringer for a mapping from the corresponding functionality on a manifold with boundary. In§ four. 6 of a similar bankruptcy we introduce the concept that of a versal deformation of a functionality with a nonisolated singularity within the dass of capabilities whose severe units are arbitrary whole intersections of mounted dimension.
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Additional resources for Dynamical Systems VIII: Singularity Theory II. Applications
The multiplicity of an edge joining two vertices is the same as for boundary singularities: it is equal to the intersection index of the corresponding cycles if at least one of them is short, and to half this index if both these cycles are long. The edge is oriented so that the intersection index will be positive. An edge that joins vertices corresponding to cycles of different lengths is equipped with a sign > that is directed towards the vertex corresponding to the short cycle. If the graph is a tree, then the orientation of edge is not indicated (it can be done arbitrarily taking account of the orientations of the cycles of the basis).
Chapter I. Classification of Functions and Mappings 36 Corollary (Aleksandrov (1985)). P(T/; z) = I zc p=1 L ll+l2+s+l=hp (m- ~ +2 /1)(1 + m lz)z-<2+l2JdPp Zd 1 - 1) ZdPp+l - 1 ZdPp-1 - 1 1 Zd 1 d -, ... ,-d--d- · 'd d ·Dl,p-h-1 ( -d--d , ... , d Z Pp Z I Z Pp - Z Pp-1 Z Pp - Z Pp+ I Z Pp Z I p As was already noted, 11 = r for quasihomogeneaus complete intersections. Corollary. 11 = P(1j_l; 1). Example. l. Q1, q = 1, ... Qd:-;; 1 )). Since for q = 1, ... Q1) are 11-dimensional (Greuel (1980)), the indicated polynomials give other formulas for the computation ofthe Milnor number.
4. Projections onto the Line. We consider a projection V where V is a complete intersection (perhaps 0-dimensional). c:.. CC"+ 1 --+ CCl, 1°. Classification Theorem (Goryunov (1981, 1982)). 1) Any simple projection of a complete intersection onto the complex line is stably equivalent to the projection of either a hypersurface or a curve in a three-dimensional space, or of a multiple point on a plane or in a three-dimensional space. ;;:. ;;:. 4 E6 E? ;;:. ;;:. 3 u B~, F4 u + x~ + xi + q u~ + xf + q ux 1 + xf + q u 2 + x~ + q Here q is a sum of squares of additional variables X;.