Algebraic solutions of ODE using padic numbers  Katz N.M.
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Название: 
Algebraic solutions of ODE using padic numbers 
Автор: 
Katz N.M. 
Категория: 
Математика

Тип: 
Книга 
Дата: 
30.12.2008 15:40:11 
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48 
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Описание: 
This paper grew out of an attempt to answer the following question, first raised by Grothendieck. Consider a linear homogeneous nxn system of firstorder differential equations
L(Y)=A{z)Y
az
in which A{z). is an n x n matrix of rational functions of z. To fix ideas, suppose that the coefficients of the entries of A(z) all lie in an algebraic number field K. Then for almost all primes p of K, it makes sense to reduce this equation modulo p, obtaining a differential equation over
F,(z).
(I) Suppose that for almost all primes p, the reduced equation has a full set of solutions (i.e., has n solutions in (F,(z))" which are linearly independent over Fq(z)). Does the original equation admit a full set of solutions in algebraic functions of z?
For example, the equation
d /л l
with aeZ may be reduced modulo p for all those primes p not dividing a, and the reduced equation admits the solution z* for any integer b such that absl modulo p. The original equation has for its solution the function z11". Of course, (I) may be reformulated in greater apparent generality. Let R be a subring of С which, as a ring, is finitely generated over Z. Let S be a smooth Rscheme with geometrically connected fibres, and consider a differential equation (M, V) on S/R, by which we understand a locally free sheaf M on S of finite rank together with an Rlinear integrable connection V: M* Qls/R®M. (We considered above the case K = 0[l/n], & the ring of integers in an algebraic number field, S an open set in P},, М = Й£, and V: M*Q\jR®M given by
Vm = dmdz®A(z)m.)
I Inventioncs math., Vol. 18 
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