"Differential Galois theory"의 두 판 사이의 차이

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<h5>introduction</h5>
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* differential galois theory
 
* differential galois theory
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* Liouville 
 
* Liouville 
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* [[2008년 12월~09년 1월 한국 방문|2008년 12월]] 9일 MCF 'differential Galois theory'
  
 
 
 
 
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* framework for linear differential equation
 
* framework for linear differential equation
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* field extension is made by including solutions of DE to the base field (e.g. rational function field)
 
*  consider monic differential equations over a differential field F<br><math>\mathcal{L}[Y]=Y^{(n)}+a_{n-1}Y^{(n-1)}+\cdots+a_{1}Y^{(1)}+a_0</math>, <math>a_i\in F</math><br>
 
*  consider monic differential equations over a differential field F<br><math>\mathcal{L}[Y]=Y^{(n)}+a_{n-1}Y^{(n-1)}+\cdots+a_{1}Y^{(1)}+a_0</math>, <math>a_i\in F</math><br>
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* made by including solutions of DE to the base field (e.g. rational function field)
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* <math>(E,\partial_E)\supseteq (F,\partial_F)</math> is a Picard-Vessiot extension for <math>\mathcal{L}</math> if<br>
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** E/F is generated by n linear independent solution to <math>\mathcal{L}</math>, i.e. adjoining basis of <math>V=\mathcal{L}^{-1}(0)</math> to F
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** <math>C_E=C_F</math>, <math>\partial_E\mid_F=\partial_F</math>
 
* this corresponds to the concept of the splitting fields(or Galois extensions)
 
* this corresponds to the concept of the splitting fields(or Galois extensions)
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* we can define a Galois group for a linear differential equation.
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* we can define a Galois group for a linear differential equation<br>  <br>
 
*  examples<br>
 
*  examples<br>
 
** algebraic extension
 
** algebraic extension
 
** adjoining an integral
 
** adjoining an integral
 
** adjoining the exponential of an integral
 
** adjoining the exponential of an integral
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theorem
 
theorem

2012년 8월 20일 (월) 23:16 판

introduction
  • differential galois theory
  • Liouville 
  • 2008년 12월 9일 MCF 'differential Galois theory'

 

 

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

 

 

solution by quadrature
  • 일계 선형미분방정식
    \(\frac{dy}{dx}+a(x)y=b(x)\)
    \(y(x)e^{\int a(x)\,dx}=\int b(x)e^{\int a(x)\,dx} \,dx+C\)
  • \(y''-2xy'=0\)
    \(y=\int e^{x^2}\, dx\)
  • note that the integral of an exponential naturally shows up in expression solutions

 

 

differential field
  • a pair \((F,\partial)\) such that
    • \(\partial(a+b)=\partial a+\partial b\)
    • \(\partial(ab)=(\partial a)b+a(\partial b)\)
  • \(C_F=\ker \partial\)

 

 

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
  • it is allowed to take exponentials and logarithms to make a field extension
  • elementary element
  • difference between Liouville extension
    • exponential+ integral <=> differentiation + exponential of integral
    • in elementary extension, we are not allowed to get an integrated 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
  • field extension is made by including solutions of DE to the base field (e.g. rational function field)
  • consider monic differential equations over a differential field F
    \(\mathcal{L}[Y]=Y^{(n)}+a_{n-1}Y^{(n-1)}+\cdots+a_{1}Y^{(1)}+a_0\), \(a_i\in F\)
  • \((E,\partial_E)\supseteq (F,\partial_F)\) is a Picard-Vessiot extension for \(\mathcal{L}\) if
    • E/F is generated by n linear independent solution to \(\mathcal{L}\), i.e. adjoining basis of \(V=\mathcal{L}^{-1}(0)\) to F
    • \(C_E=C_F\), \(\partial_E\mid_F=\partial_F\)
  • 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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