Singularity Theory I
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Springer-Verlag Berlin Heidelberg GmbH
V. 1. Arnold V. V. Goryunov O. V. Lyashko V. A.Vasil'ev
Singularity Theory 1
.~.
~
Springer
Consulting Editors of the Series: A.A. Agrachev, A.A.Gonchar, E.F. Mishchenko, N.M. Ostianu, V. P. Sakharova, A.B. Zhishchenko Title of the Russian edition: Itogi nauki i tekhniki, Sovremennye problemy matematiki, Fundamental'nye napravleniya, VoI. 6, Dinamicheskie sistemy 6 Publisher VINITI, Moscow 1988
Second Printing 1998 ofthe First Edition 1993, which was original1y published as Dynamical Systems VI, Volume 6 of the Encydopaedia of Mathematical SCÎences.
Die Dcut5che Bibliothek - Clp·EinbeitsaufDabme
Sinplarity lbro'1 / by v. 1. Amal'd. ... - Berlin ; Heid.elberg ; New York; Barcelonl ; BudapeSl ; HODpollg ; London ; Mailand. ; Paris; SaDla Oara; Siogapur ; To~o : Springer 1. - 1. ed.., 2. prinling. - 1998 (Eocyclopacd.ia of DlIlbc:lIlllicll SCkDCeI ; \tiI. 6) ISBN 978-3-540-63711-0 ISBN 978-3-642-58009-3 (eBook) DOI 10.1007/978-3-642-58009-3
Mathematics Subject Classification (1991): Primary 58C27, Secondary 14805, 14E15, 32S05, 32SXX, 58C28
ISBN 978-3-540-63711-0
Thi. work Îllubj«t to COpyrighL AI! righu are rtlerved, wbetbel the whole OI part of Ihe material ÎI ooncemed., ' pKÎfically the lightl of IRnslltioD, rtprinting, leUle of iIIwtrltions, leatltion, broadClolting, leproduction on microliJms OI in Iny othel way, and ltonge in dati banks. Duplication of thil publication or parti thereaf is per· mitled. only under the provisionl of the German Copyrightl.aw of Stptembtr 9, 1965, in ilS currtnt ~nion, and ~rmi$$ion for IlS 0, ao ~ 0 4d 3 + 27 ~ 0 p>O,ao~O
a~ ~
4
p > 0, ao
~
0
q>O,ao~O q>O,ao~O
15 + 2q - 1 15 + 2q
4 series of singularities of corank 3: Name
Normal form
Q2.0 Q2,p 51.0 51.p
x 3 + yz2 + ax 2y2 + xy4 x 3 + yz2 + x 2i + ay6+p x 2z + yz2 + l + azy3 2 x z + yz2 + X2y 2 + ayHP x 2z + yz2 + zl + axy3+Q x 2z + yz2 + zy3 + ax 2yl+Q x 3 + xz 2 + xy3 + ay3z x 3 + xz 2 + xy3 + ai +Qz2 x3 + xz 2 + xy3 + ay3+ Qz
5:' 2q- 1 5:' 2Q UI.O
UI.2Q-1 UI.2Q
Restrictions a~ ~
4
p>O,ao;CO a~
;c 4
p>O,ao;CO q>O,ao~O
q>O,ao;cO ao(a~ + I) ;c 0 q > 0, ao ;c 0 q>O,ao;cO
Multiplicity II
14 14 + P 14 14 + P
14 + 2q - 1 14 + 2q 14 14 + 2q - 1 14 + 2q
14 exceptional families: E I8
£19 £20 ZI7 ZI8 ZI9 U I6
x 3 + ylO + axy7 x 3 + xy7 + ayll x 3 + i l + axy8 x 3y + y8 + axy6 x 3y + xy6 + ay9 x 3y + y9 + axy7 3 x + xz 2 + yS + ax2i
W17
WI8
QI6 QI7 QI8 5 16 517
x4+xyS+ay7 x4 + y7 + ax2y4 x 3 + yz2 + y7 + axyS x 3 + yz2 + xyS + ay8 x 3 + yz2 + y8 + axy6 x 2z + yz2 + xy4 + ay6 x 2z + yz2 + y6 + azy4
In the last three tables a = a o + a1y. 2.4. Simple Singularities and Klein Singularities. A remarkable connection exists between the classification of simple singularities, that of regular polyhedra in three-dimensional space, and that of the Coxeter groups A k , Dk , E k • Although the coincidence of these classifications emerges only when classification theo-
26
Chapter I. Critical Points of Functions
rems proved independently in the three indicated settings
