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

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

100 which this form is

100 which this form is defined (see e.g. the discussion in [MvS] or [W] ). However, our assumptions insure that it is defined on all branches. Also, as F 0 is equivariant, det(dxF0) is invariant, as is dX. Thus, co is G-invariant, so L o is a trivial representation. [] Corollary 3 If wt(~,) is odd, then s b is not an integer. Hence, ~(F0)sb = 0 so by theorem 1, Xb = 0. A similar agument works if s is odd since then s d is not an integer. 0 Corollary 4 Corollary 5 This is just a restatement of corollary 2 of [I)4]. This follows by applying the preceding results and corollaries to F = gradxH, and observing that: Xc for H---Xp for F, Xe and 7,0 for H = Xhd + and Xhd- for F, and Xc + for H = Xhb + for F. o Proofs of the Results for the Reduced Methods Corollary 6 If we let X.~ + denote the ordinary characters for the representations appearing in the decomposition of J(F0)sp given by proposition 1.2 (there is no V 0 in this case). Then Hence, sigG¢(5(F0)sp) = X~l; + - X.~- = 8G(Xj¢ ;) = 8G(sigGc;(J(F0)sp)) = sigG(.\$(F0)sp) - Xp-1 - 8G(EXie) by theorem 1. o Lemma 3.5 X g~ - 2'X5¢;- • mod 2..7C(G) For .9(F0) it is enough to see that all n x n minors have weight > Sp. In fact, the weight of an nxn minor= i=t ~ di -( j=~l aj+ wt(X))+c = where c = wt(~.) or some a i. Thus, c > 0; and J(F0)sp -- I(F0)sp. For ~'Fo) we examine instead det(dxFO). wt(det(dxFO)) = ~ d i-~ aj = s > i=i j=1 yielding the desired conclusion. 0 s= 1/2.wt(~,) = s b p+c

Corollary 7 101 For i) we know by theorem 1 and lemma 3.5, number ofnontrivialbranches ~ dim[! .~(F0)sp ~ dim[IQ(F0)sp mod2. For ii) we use the decomposition B' = B+ u B_, lemma 1.1 and lemma 3.5 to conclude (6.3) Also, (6.4) 1/2(card(B+) - card(B_)) -- ~b(1) - dim~ ~(F0)sb b~) mod 2. 1/2(card(B3) --- j(F)+ 1 rood2 Adding card(B_) to (6.3) and using (6.4) we obtain (6.5) j(F) + 1 m b(F) + card(B) mod2 m dim~Q(F0)sb mod 2 Since card(B+) ~ card(B) mod 2, the result follows from (6.5). Then iii) follows by a similar argument using instead the decomposition B' -- B e U B o. [] Corollary 8 If (j(F), b(F), d(F)-l) ~ 0 6 (1/2E) 3 then we wish to conclude that there exist nontrivial branches. First, if j~) # 0 then by i) of corollary 7, there is a nontrivial branch. Next, suppose that j(F) -- 0 but bfF) # 0. Then, by ii) of corollary 7, card(B+) --- 0 mod 2. Since there is the trivial half-branch in B+, there must be another. Lastly, if j(F) - b(F) -- 0, but d(F)-I # 0 mod 2, then d(F) ---- 0 mod 2. If there is only the trivial branch, then deg(f) can be computed by using F(x,2q3) for k 0 sufficiently small and close to 0 so we obtain deg(f) = sign(det(dxF(0, ~))) for 2q3 on either side of 0. Thus, the signs must be the same and card(Be) is even, contradicting iii) of corollary 7. Alternatively we could use that d(F) is the mod 2 degree of f; similar reasoning to the above would imply that it is both even and odd, again a contradiction, tJ §7 Proofs of the Formulas for the Bifurcation Character We shall prove the formulas for the bifurcation characters and see how the proof of I) of theorem 2 is related to that for the bifurcation character. We begin by attempting to determine the analogue of the Jacobian algebra for bifurcation and concluding that it is the bifurcation algebra.

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