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Differential modules of bounded spectral norm. (English) Zbl 0765.12003

p-adic methods in number theory and algebraic geometry, Contemp. Math. 133, 39-58 (1992).
[For the entire collection see Zbl 0752.00052.]
This paper deals with a differential module \((M,\nabla)\) of finite dimension \(n\) over the field \(E\) of \(p\)-adic analytic elements defined over the complete \(p\)-adic field \(K\). The material developed here has an application in the determination of effective \(p\)-adic bounds for the holomorphic part of the solution matrix of a differential system at a regular singular point. A basis \(e\) on \(M\) over \(E\) naturally determines a norm \(|\cdot| _ e\) on \(M\). It is known that the spectral norm of \(\delta = \nabla (d/dx)\), \(| \delta| _{sp} = \lim_ s| \delta^ s| _ e^{1/s}\) (independent of the chosen norm \(|\cdot| _ e\)) is bounded by 1 iff the horizontal sections of \(M\) at a generic point \(t\) converge in the disk \(D(t, (p^{1/(p-1)})^ -)\). The problem is to decide whether assuming \(| \delta| _{sp} \leq 1\) and given a basis \(e\) of \(M\), one can determine a change of basis \(e' = eH\), \(H \in GL(n,E)\), in such a way that \(| \delta| _{e'}\leq 1\) \(| H| | H^{-1}| \leq \sup (1, | \delta| _ e)^{n-1}\).
This problem, Conjecture A of the paper, is here solved in the positive sense if \(n = 3\). (The case \(n =2\) had been explained by the authors in a previous paper).
Besides analysing this important special case, the authors nicely summarize some of their previous general results to also provide a treatment of the case \(| \delta| _{_ sp} > 1\).

MSC:

12H25 \(p\)-adic differential equations
11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)

Citations:

Zbl 0752.00052
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