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40 Chapter 2. Global aspects and
R(eiφ[λνk +···+λνk +λνqμ +λνk +···+λνk])<R(eiφζ(ak)), 1 p−1 p+1 n
for 0 < φ < ε0. This contradicts the assumption that C ∈ Gζ(ak) and so (ii) is proved.
We are now ready to prove some results concerning the structure of the set of solutions Γ of ARE.
Theorem 2.4.3. Let the eigenvalues λ1 , . . . , λm+n of M be simple. Then (i) Γk∩Γj =∅forallk̸=j,1≤k,j≤N.
k {k} if v˜(ak)̸=0, (ii) J(a )= ∅ else.
(iii) There exist at most N solutions Wk = (wk )1≤l≤m of ARE which are given
as
lα 1≤α≤n wk = vlα(ak), k ∈ J(ak).
lα v˜(ak )
Proof. From Lemma 2.4.2, (ii), since in the case of simple eigenvalues
(2.34)
(λν1k,λν2k,...,λνnk)̸=(λν1j,...,λνnj )
for j ̸= k, we infer for all C ∈ Gζ(ak) and j ̸= k, D(aj,C) = 0.
This proves (i) and also (ii) since J(ak) contains those indices such that C ∈ G0ζ(ak). (iii) then follows from the representation formula (2.33) where only one C-determinant, namely D(ak,C), is non-zero and cancels out.
It is clear that in the case of multiple eigenvalues (notice that here we as- sumed that they are all of full geometric multiplicity) the sets J(ak) may contain more than one element. Therefore we describe the structure of Γ by the number of elements in J(ak) which can be determined from the set of eigenvalues and their multiplicity.