"Differential Galois theory"의 두 판 사이의 차이

2012년 8월 20일 (월) 22:48 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
- exponentials of integrals
- algebraic extension (generalized Liouville 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
- \(e_i'\in K_{i-1}\), i.e. \(e_i=\int e_i'\in K_i\)
- \(e_{i}'/e_{i}\in K_{i-1}\) i.e. \((\log e_i)' \in K_{i-1}\)
- \(e_{i}\) is algebraic over \(K_{i-1}\)
remark on exponentiation
- Let \(a,a'\in F\). Is \(b=e^a\in K\) where K is a Liouville extension?
- \(b'=a' e^a=a'b\) implies \(a'=\frac{b'}{b}\in F\).
- therefore
a few result
- K/F is a Liouville extension iff the differential Galois group K over F is solvable.
- K/F is a generalized Liouville extension iff the differential Galois group K over F has the solvable component of the identity

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

@@ 52번째 줄: / 52번째 줄: @@
 * an element is said to be representable by a generalized quadrature
 *  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>, i.e.<br>
+** <math>e_i'\in K_{i-1}</math>, i.e. <math>e_i=\int e_i'\in K_i</math><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>
-*  K/F is a Liouville extension iff the differential Galois group K over F is solvable.<br>
+*  remark on exponentiation<br>
-*  K/F is a generalized Liouville extension iff the differential Galois group K over F has the solvable component of the identity<br>
+**  Let <math>a,a'\in F</math>. Is <math>b=e^a\in K</math> where K is a Liouville extension?<br>
+** <math>b'=a' e^a=a'b</math> implies <math>a'=\frac{b'}{b}\in F</math>.<br>
+**  therefore<br>
+*  a few result<br>
+**  K/F is a Liouville extension iff the differential Galois group K over F is solvable.<br>
+**  K/F is a generalized Liouville extension iff the differential Galois group K over F has the solvable component of the identity<br>