Differential Galois theory

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)
• examples
  
  • algebraic extension
  • adjoining an integral
  • adjoining the exponential of an integral
• we can define a Galois group for a linear differential equation
  \(\operatorname{Gal}(E/F)=\{\sigma\in\operatorname{Aut}E|\partial(\sigma(n))=\sigma(\partial a), \sigma\mid _F=\operatorname{id} \}\)
  
  • the action of an element of the Galois group is determined by its action on a basis of V

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

• Differential Galois Theory and Non-Integrability of Hamiltonian Systems
• Integrability and non-integrability in Hamiltonian mechanics
• [1]http://www.caminos.upm.es/matematicas/morales%20ruiz/libroFSB.pdf
• http://andromeda.rutgers.edu/~liguo/DARTIII/Presentations/Khovanskii.pdf
• http://www.math.purdue.edu/~agabriel/topological_galois.pdf

 

 

related items

• Class Field Theory
  
• number fields and threefolds
  
• 틀:수학노트

 

 

 

encyclopedia

• http://ko.wikipedia.org/wiki/
• http://en.wikipedia.org/wiki/Differential_Galois_theory
• http://en.wikipedia.org/wiki/Homotopy_lifting_property
• http://en.wikipedia.org/wiki/covering_space
• http://en.wikipedia.org/wiki/Field_extension

 

 

articles

• Crespo, Teresa, Zbigniew Hajto, and Elzbieta Sowa-Adamus. ‘Galois Correspondence Theorem for Picard-Vessiot Extensions’. arXiv:1502.08026 [math], 27 February 2015. http://arxiv.org/abs/1502.08026.
• Singer, Michael F. ‘Liouvillian First Integrals of Differential Equations’. Transactions of the American Mathematical Society 333, no. 2 (1 October 1992): 673–88. doi:10.2307/2154053.

books

• Group Theory and Differential Equations
  
  • Lawrence Markus, 1960
• An introduction to differential algebra
  
  • Irving Kaplansky
    
• algebraic theory of differential equations
  
• http://gigapedia.info/1/galois_theory
• http://gigapedia.info/1/differential+galois+theory
• http://gigapedia.info/1/Kolchin
• http://gigapedia.info/1/ritt
• http://gigapedia.info/1/Galois'+dream
• http://gigapedia.info/1/differntial+algebra