By Donald G. Babbitt
ISBN-10: 0821823264
ISBN-13: 9780821823262
Read Online or Download Deformations of Nilpotent Matrices over Rings and Reduction of Analytic Families of Meromorphic Differential Equations PDF
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Additional resources for Deformations of Nilpotent Matrices over Rings and Reduction of Analytic Families of Meromorphic Differential Equations
Example text
K. We continue to work with a Noetherian and integrally closed R and a finite torsion free R-module V_ which is reflexive. We shall now K introduce a partial order in the set of nonzero nilpotents in L . If L,M € ^ ( V _ ) \ (0), we write R if either the dimension of the GL(Y»)-orbit of M is strictly greater than the dimension of the GL(Y^)-orbit of L, or else, L and M are conjugate under GL(V V) and XV £(M) < ^i(L) . m Here < is the natural lexicographic ordering in Z 1 , m. 8. Clearly the relation L « M depends only on the GL(V_)-orbits R of L and M.
S . VARADARAJAN so that T = N\N-.. It is a question o f proving that if N is sufficiently small, N 1 t H L(t) is open. Let u s choose n o w N sufficiently small so that is a nilpotent deformation o f L ( t 1 ) for each in (a) v y then show that if t n e N n , I 1 1 m.. x(t) ' = Im.. , N.. is open. t e N . The results for t in a neighborhood Lemma 2. Suppose the deformation t H L(t) is such that i (t) = I m v(t„) for all t in a neighborhood of t„. Then t H W-m_ _ v(t) y 0 ' and n 0 t H W 1 (t) are regular families for t varying in a neighborhood of t .
Contain component of N itself when the t . Applying the induction hypothesis to each irreducible T Suppose then that r we can conclude by contracting for all t e N 0 T±. i° = I (t_). Then m m.. 0 for a11 t e N " V' x A)/W n(t R = Wm,-x o " -m,(trt 0), If L t '( ^ N still further that Suppose first that / < &m (t ). and so we have (k) for all t e N. t e NX^ view of Lemma 3 we may assume that W (5) T so that (a) and (b) of Lemma 1 are true, By Then £(L(t)) < £(L(t )) for m -1^0^ and even for if necessary we may assume that all the irreducible components ^(L(t)) ^£(L(t 0 )) W dim(T) and the spread I m = l (t ) for all t e N.
Deformations of Nilpotent Matrices over Rings and Reduction of Analytic Families of Meromorphic Differential Equations by Donald G. Babbitt
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