"Differential Galois theory"의 두 판 사이의 차이

2010년 8월 13일 (금) 15:51 판

integration in finite terms
quadrature of second order differential equation (Fuchsian differential equation)

basic functions : basic elementary functions
allowed operatrions : compositions, arithmetic operations, differentiation, integration
examples
- an elliptic integral is representable by quadrature

we can adjoin integrals and exponentials of integrals + algbraic extension
an element is said to be representable by a generalized quadrature
For\(K_{i}=K_{i-1}(e_i)\) , one of the following condition holds
- \(e_i'\in K_{i-1}\)
- \(e_{i}'/e_{i}\in K_{i-1}\) i.e. \((\log e_i)' \in K_{i-1}\)
- \(e_{i}\) is algebraic over \(K_{i-1}\)

framework for linear differential equation
made by including solutions of DE to the base field (e.g. rational function field)
this corresponds to the concept of the splitting fields(or Galois extensions)
we can define a Galois group for a linear differential equation.
examples
- algebraic extension
- adjoining an integral
- adjoining the exponential of an integral

theorem

If a Picard-Vessiot extension is a Liouville extension, then the Galois group of this extension is solvable.

theorem

A Fuchsian linear differential equation is solvable by quadratures if and only if the monodromy group of this equation is solvable.

Liouvillian First Integrals of Differential Equations
- Michael F. Singer, Transactions of the American Mathematical Society, Vol. 333, No. 2 (Oct., 1992), pp. 673-688
Elementary and Liouvillian solutions of linear differential equations
- M. F. Singer and J. H. Davenport, 1985

@@ 50번째 줄: / 50번째 줄: @@
 *  For<math>K_{i}=K_{i-1}(e_i)</math> , one of the following condition holds<br>
 ** <math>e_i'\in K_{i-1}</math><br>
-** <math>e_{i}'/e_{i}\in K_{i-1}</math> i.e. <br>
+** <math>e_{i}'/e_{i}\in K_{i-1}</math> i.e. <math>(\log e_i)' \in K_{i-1}</math><br>
 ** <math>e_{i}</math> is algebraic over <math>K_{i-1}</math><br>
+*   <br>
@@ 109번째 줄: / 110번째 줄: @@
 * [[Class Field Theory]]<br>
 * [[number fields and threefolds]]<br>
+* [http://pythagoras0.springnote.com/pages/6233973 Fuchsian 미분방정식(Fuchsian differential equation)]<br>
@@ 132번째 줄: / 134번째 줄: @@
 * [http://www.jstor.org/stable/2154053 Liouvillian First Integrals of Differential Equations]<br>
 ** Michael F. Singer, Transactions of the American Mathematical Society, Vol. 333, No. 2 (Oct., 1992), pp. 673-688
+* [http://dx.doi.org/10.1007/3-540-15984-3_335 Elementary and Liouvillian solutions of linear differential equations]<br>
+** M. F. Singer and J. H. Davenport, 1985