The Inverse Problem in Di erential Galois Theory
The Inverse Problem in Di erential Galois Theory
复制标题
微分伽罗瓦理论中的反问题
DOI：
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发表时间：
1996
期刊：
影响因子：
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通讯作者：
M. Singer
中科院分区：
文献类型：
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作者：
M. Singer
1.伽罗瓦理论通常的伽罗瓦多项式方程理论允许人们以这样一种方式将群与多项式联系起来，即多项式根的代数性质反映在群的性质中。更具体地说，给定一个域k和一个多项式p(X)，其中p(X)的所有根都形式地邻接到k，从而形成p(X)的分裂域K.伽罗瓦群G被定义为K的所有自同构群，使得k的每个元素都x.对于线性微分方程组也可以进行类似的构造。一开始是一个不同的eldk。这是一个eld1和一个映射0：k！K(称为导数)满足两个性质：(a+b)0=a0+b0和(Ab)0=a0 b+ab0对所有a；b2k。可微域的例子有复数C的ELD、有理函数C(X)的ELD、形式幂级数环的商ELD C((X))和收敛的幂级数环的商ELD C(FXG)。子带Ck=Fc 2 k j c 0=0g称为k的常量的子带(当上下文清楚时，我们将放弃这是第二作者在1995年5月31日至6月3日在格罗宁根举行的斯托克斯现象会议上发表的讲话的略微扩大的版本。作者对组织者的邀请表示感谢。1本文中的所有域都假定为特征零。
1. Galois Theory The usual Galois theory of polynomial equations allows one to associate a group to a polynomial in such a way that the algebraic properties of the roots of the polynomial are reeected in properties of the group. More speciically, given a eld k and a polynomial p(x) with coeecients in the eld, one forms the splitting eld K of p(x) by formally adjoining all the roots of p(x) to k. The Galois group G is deened to be the group of all automorphisms of K that leave each element of k xed. A similar construction can be made for linear diierential equations. One starts with a diierential eld k. This is a eld 1 together with a map 0 : k ! k (called a derivation) satisfying the two properties: (a + b) 0 = a 0 + b 0 and (ab) 0 = a 0 b + ab 0 for all a; b 2 k. Examples of diierential elds are the eld of complex numbers C, the eld of rational functions C(x), the quotient eld C((x)) of the ring of formal power series and the quotient eld C(fxg) of the ring of convergent power series. The subbeld C k = fc 2 k j c 0 = 0g is called the subbeld of constants of k (when the context is clear we shall drop This is a slightly enlarged version of a talk given by the second author at the Stokes Phenomenon Conference held in Groningen, May 31-June 3, 1995. The authors would like to thank the organizers for inviting them. 1 All elds in this paper will be assumed to be of characteristic zero.