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SN~ (~6) lff 2It 3k_~ , 5 ",,x_J {b)

SN~ (~6) lff 2It 3k_~ , 5 ",,x_J {b)

104 form is clearly

104 form is clearly positive on all branches and G-invariant. Then -- ~,.Jac(F).dX/Jac(F) = ~..Jac(F)-o, R~- = 0 and R~+ ,,, A/(~,.Jac(F)).{co} = Q(F1).(¢o). Arguing as before, the residue pairing on Ro~+ is given by the multiplication pairing on Q(F 1) composed with the linear functional As above, Jac(F1).(o = d(L.Jac(F)) res"(h) = res(Jac(F1).(~,.Jae(F))- 1.c0 ) = Res(h.(k.Jac(F)) -1.c0). res( (~..Jac(F))- 1.Jac(F1).¢o) res((~,.Jac(F))-l.d(~,.Jac(F))) > 0, the linear functional is positive on JacfF1) as required. [] AFN AFN2 Ch C D1 D2 Bibliography Aoki,K., Fukuda, T. and Nishimura T. On the number of branches of the zero locus of a map germ (~qn,0) ----* (IRn-l,0), Topology and Computer Sciences, S. Suzuki ed. Kinokuniya Co. Ltd. Tokyo (1987) 347-367 Aoki,K., Fukuda, T. and Nishimura T. An algebraic forrauia for the topological types of one parameter bifurcation diagrams, Arch. Rat'l. Mech. and Anal. 108 (1989) 247-266 Chossat, P. Solutions avec sym6trie di6drale dans les probldmes de bifurcation invariants par sym6trie sph6rique, Comptes Rendus ser 1 297 (1983) 639-642 Cicogna, G. Symmetry breakdown from bifurcations, Letters al Nuovo Cimento 31 (1981) 600-602 Damon, J. Topological triviality and versality for subgroups of A and K 1I: Sufficient conditions and applications, preprint Topological equivalence of bifurcation problems, Nonlinearity 1 (1988) 311-331

105 D3 On the number of branches for real and complex weighted homogeneous curve singularities, to appear Topology D4 G-signature, G-degree, and the symmetries of branches of curve singularities, to appear Topology DP Dixon, J and Puttawamaiah, B.M. Modular Representations of Finite Groups, Pure and Applied Math. Series Academic Press (1977) New York- London EL Eisenbud, D. and Levine, H. An algebraic formula for the degree of a C~-map germ, Annals of Math. 106 (1977) 19-44 F Field, M. Symmetry breaking for the Weft groups W(D~, preprint FR Field, M. and Richardson, R.W. Symmetry Breaking and the maximal isotropy subgroup conjecture for reflection groups, Arch. Rat'1. Mech. and Anal. 105 (1989) 61-94 G Golubitsky, M. The Benarcl Problem, symmetry and the lattice ofisotropy subgroups, in Bifurcation Theory, Mechanics and Physics, C. P. Bruter et aL eds., D. Reidel, Dordrecht-Boston-Lancaster (1983) 225-257 GZ Guscin-Zade, S. M. An F_,quivatiant Analogue of the index of a gradient Vector Field, Springer Lecture Notes 1216 (1986) 196-210 L Lauterbach,R. An example of symmetry breaking with submaximal isotropy, Contemp. Math. 56 (1986) 217-222 MvS Monuddi, J. and van Straten, D. One-forms on singular curves and the topology of real curves singularities ,to appear Topology R Roberts, M. E, quivafiantMilnornumbers andinvafiantMorse approximations, J. London Math. Soc. (2) 31 (1985) 487-500 S Serre, J.P. Linear Representations of Finite Groups, Graduate Texts in Math. Springer-Verlag (19) Heidelberg-New York

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