"Differential Galois theory"의 두 판 사이의 차이

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37번째 줄: 37번째 줄:
 
<h5>elementary extension</h5>
 
<h5>elementary extension</h5>
  
−
* using exponential and logarithm
+
* to takitaking exponentials and logarithms
 
* elementary element
 
* elementary element
  
46번째 줄: 46번째 줄:
 
<h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">Liouville extension</h5>
 
<h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">Liouville extension</h5>
  
−
*  we can adjoin<br>
+
* an element is said to be representable by a generalized quadrature
 +
* we can capture these properties using the concept of Liouville extension
 +
*  to get a Liouville extension, we can adjoin<br>
 
** integrals
 
** integrals
 
** exponentials of integrals
 
** exponentials of integrals
−
** algebraic extension (generalized Liouville extension)
+
** algebraic extension (generalized Liouville extension)<br>
−
* an element is said to be representable by a generalized quadrature
+
***  from these we can include the following operations<br>
 +
**** exponential
 +
**** logarithm
 
*  For<math>K_{i}=K_{i-1}(e_i)</math> , one of the following condition holds<br>
 
*  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. <math>e_i=\int e_i'\in K_i</math><br>
 
** <math>e_i'\in K_{i-1}</math>, i.e. <math>e_i=\int e_i'\in K_i</math><br>
58번째 줄: 62번째 줄:
 
**  Let <math>a,a'\in F</math>. Is <math>b=e^a\in K</math> where K is a Liouville extension?<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>
 
** <math>b'=a' e^a=a'b</math> implies <math>a'=\frac{b'}{b}\in F</math>.<br>
−
**  therefore<br>
+
**  the exponential of the integral of a' i.e. <math>e^{\int a'}=e^a+c</math> must be in the Liouville extension. So <math>b=e^a\in K</math>.<br>
 +
*  remark on logarithm<br>
 +
** <math>b=\log a</math> is the integral of <math>a'/a\in F</math>. So <math>b\in K</math><br>
 +
 
 
*  a few result<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 Liouville extension iff the differential Galois group K over F is solvable.<br>

2012년 8월 20일 (월) 22:55 판

  • differential galois theory
  • Liouville 

 

 

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

 

 

differential field
  •  

 

 

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

 

 

elementary extension
  • to takitaking exponentials and logarithms
  • elementary element

 

 

Liouville extension
  • an element is said to be representable by a generalized quadrature
  • we can capture these properties using the concept of Liouville extension
  • to get a Liouville extension, we can adjoin
    • integrals
    • exponentials of integrals
    • algebraic extension (generalized Liouville extension)
      • from these we can include the following operations
        • exponential
        • logarithm
  • 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 \(a'=\frac{b'}{b}\in F\).
    • the exponential of the integral of a' i.e. \(e^{\int a'}=e^a+c\) must be in the Liouville extension. So \(b=e^a\in K\).
  • remark on logarithm
    • \(b=\log a\) is the integral of \(a'/a\in F\). So \(b\in K\)
  • 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

 

 

Picard-Vessiot extension
  • 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.

 

 

Fuchsian differential equation
  • differential equation with regular singularities
  • indicial equation
    \(x(x-1)+px+q=0\)

theorem

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

 

 

 

solution by quadrature

 

 

related items

 

 

 

encyclopedia

 

 

articles

 

 

books
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