# Dynamical Systems VIII: Singularity Theory II. Applications by V.I. Arnol'd, J.S. Joel, V.V. Goryunov, O.V. Lyashko, V.A. PDF

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 innovations and strategies of the speculation of singularities of delicate mappings and features. This idea has a number of purposes in arithmetic and physics; right here we commence describing those applica tions. however the current quantity is largely self sustaining of the 1st one: the entire strategies of singularity concept that we use are brought during the presentation, and references to EMS 6 are restrained to the quotation of technical effects. even supposing our major objective is the presentation of analready formulated concept, the readerwill additionally come across a few relatively fresh effects, it seems that 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 straightforward singularities, supply the definition of the Mond invariant N within the spirit of "hunting for invariants", and we draw the reader's realization to a mode of making clone of a mapping from the corresponding functionality on a manifold with boundary. In§ four. 6 of a similar bankruptcy we introduce the idea that of a versal deformation of a functionality with a nonisolated singularity within the dass of features whose severe units are arbitrary entire intersections of mounted dimension.

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**Example text**

Remark. The pair (

We note that in perturbing a projection of a subvariety V with singularities, we consider not only the change in its position relative to the fibers of the bundle, but also we deform V itself, making it nonsingular. sl-classification of mappings of singular varieties into smooth ones. Weshall call a projection a "projection onto" if the dimension of the space V being mapped is not less than the dimension of the base B. The start of a classification of projections "onto" up to the fiber-contact group left-right classification of mappings into a manifold of smaller or equal dimension) has been obtained by Goryunov (1983b, 1984).

Oy)· We take the e; tobe quasihomogeneaus mappings. The number of these mappings that have fixed degree IY. is equal to the coefficient of za in the polynomial P(T/; z). = n- p > Theorem (Aleksandrov (1983, 1985)). L z) f' = (-1)m-1zc + zc fl (1 0. Then - z-di) fl(1- z-w,) X { -l)mzc res( t-0 + (-l)m-1 t 1 -m X (1 n (1 + tzw') + t)- 1 ~~~----,-(1 + tzdi) n o::z-w, -l::z-dj- 1] }· For concrete computations this formula is applied in a transformed form. First, following Greuel and Hamm (1978), we introduce polynomials W, and D1• W,(y1, ...